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ELEMENTS 



OF 



MACHINE DESIGN 



BY 

DEXTER S. KIMBALL, A.B. 

Professor of Machine Design and Construction, Sibley College, Cornell University. 
Formerly Works Manager, Stanley Electric Manufacturing Company- 
Member of the American Society of Mechanical Engineers. 



AND 

JOHN H'. BARR, M.S., M.M.E. 

Consulting Engineer Remington Typewriter Co. Formerly Professor of Machine 

Design, Sibley College, Cornell University. Member of the 

American Society of Mechanical Engineers. 



FIRST EDITION 
NINTH THOUSAND 



NEW YORK 
JOHN WILEY & SONS, Inc. 

London : CHAPMAN & HALL, Limited 
l 9 l 5 






Copyright 1909 

BY 

DEXTER S. KIMBALL and JOHN H. BARR 



Transfer 

Army War College 

June 20 1933 




Electrotyped and Printed by Publishers Printing Co., New York, U.S. A. 



i. PREFACE 

This book is the outgrowth of the experience of the authors 
in teaching Machine Design to engineering students in Sibley 
College, Cornell University. It presupposes a knowledge of 
Mechanism and Mechanics of Engineering. While the former 
subject is a logical part of Machine Design, it may be, and usually 
is, for convenience, treated separately and in advance of that 
portion of the subject which treats of the proportioning of machine 
parts so that they will withstand the loads applied. The same 
logical order is usually followed in actual designing, as it is, 
ordinarily, necessary and convenient to outline the mechanism 
before proportioning the various members. 

With the mechanism determined, the remainder of the work 
of designing a machine consists of two distinct parts : 

(a) Consideration oH:he energy changes in the machine, and 
the maximum forces resulting therefrom. 

(b) Proportioning the various parts to withstand these forces. 

Thjs logical procedure, and the fundamental principles under- 
lying the first part (a), are seldom made clear to the student, 
in works of this character; and such information as is given on 
energy transformation in machines is, in general, that relating 
to special cases or types. A thorough understanding of these 
general principles is, however, in most cases, essential to success^ 
ful design, since a consideration of the machine as a whole 
necessarily precedes consideration of details. A very brief 
discussion of typical energy and force problems is given, there- 
fore, in Chapter II, in the hope of making this important matter 
somewhat clearer to the beginner. 

While the treatment presented presupposes a knowledge of 
Mechanics of Materials, a brief discussion of the more important 
straining actions is given in Chapter III, partly to make the appli- 
cation of the various formulae to engineering problems somewhat 

iii 



IV PREFACE 

more definite, and partly to present such rational theory as is of 
assistance in selecting working stresses and factors of safety. 
This discussion serves also to show why certain equations have 
been selected in preference to others, and also to collect in concise 
form the more important equations relating to stress and strain 
with which the designer needs to be familiar. 

The general principles of lubrication and efficiency are 
discussed in Chapter IV. Both of these are of prime importance 
to the engineer; and while the discussion is necessarily brief it is 
believed that the fundamental principles are fully covered. 

The remainder of the book is devoted to the discussion of 
some of the more important machine details, with a view of 
showing how the theoretical considerations and equations dis- 
cussed in the first part of the work are applied and modified in 
practice. The treatise is, in no sense, a hand-book, neither is it 
a manual for the drafting room, but is a discussion of the funda- 
mental principles of design, and only such practical data have been 
collected as are needed to verify or modify logical theory. It is 
hoped that the illustrative numerical examples which are intro- 
duced throughout the work may, in conjunction with the analy- 
tical methods given, suggest proper treatment of practical prob- 
lems in design. The treatment of all topics is necessarily brief, 
as it was desired to obtain a text-book which could be conve- 
niently covered in one college year and yet present the salient 
features of the subject needed by the student as a preparation 
and basis for more advanced work. While intended primarily for 
engineering students it is hoped that it may also prove of some 
interest to the practising designer. It has been the endeavor in 
the preparation of the book not only to develop rational analytical 
treatment, with due regard to constructive considerations and 
other practical limitations, but to reduce the analysis to such 
forms and terms that definite numerical results can be obtained 
in concrete problems. 

Considerable of the matter contained in the book has already 
been published, specially for the use of students in Sibley College, 
under the title of "Special Topics on the Design of Machine 
Elements," by John H. Barr, and also in " Elements of Machine 



PREFACE V 

Design," Part I, by the Authors. The writers have availed them- 
selves freely of the work of many others in the field, for which 
due credit is given in the text. 

The authors are especially indebted to Professor G. F. Blessing 
of Swarthmore College, Professors W. N. Barnard, L. A. Darling, 
and C. D. Albert of Sibley College, Cornell University, all of 
whom have given instruction in the course at various times, and 
also to Mr. A. J. Briggs, for many helpful suggestions and 
criticisms. They will be very grateful for further suggestions 
or criticisms which will improve the book. 

D. S. K. 

J. H. B. 

Ithaca, N. Y., June, 1909. 



CONTENTS 



CHAPTER I 

PAGE 

Introductory. Definitions and Fundamental Principles 
of Machine Design, i 



CHAPTER II 

The Energy and Force Problem. Consideration of Machines 
as a Means of Modifying Energy, 6 

CHAPTER III 

Straining Actions in Machine Elements. Fundamental 
Formulas for Strength and Stiffness, . . . . 31 

CHAPTER IV 

Friction, Lubrication, and Efficiency, . . . . .96 

CHAPTER V 

Springs, .114 

CHAPTER VI 

Riveted Fastenings, . . . , . . . 136 

CHAPTER VII 

Screws and Screw Fastenings, . . , . =156 

CHAPTER VIII 

Keys, Cotters, and Force Fits, . . . . . . 190 

CHAPTER IX 

Tubes, Pipes, Flues, and Thin Pl/ptes . « . . .211 



viii CONTENTS 

CHAPTER X 



l'AGE 



Constraining Surfaces, Sliding Surfaces, Journals, Bearings, 
Roller and Ball Bearings, . . . . . . .232 

CHAPTER XI 
Axles, Shafting, and Couplings, ...... 285 

CHAPTER XII 

Belt, Rope, and Chain Transmission, .-„,,_. 309 

CHAPTER XIII 

Applications of Friction. Friction Wheels, Friction Brakes, 
and Clutches, ..,.'. 350 

CHAPTER XIV 

Toothed Gearing, Spur, Bevel and Screw Gears, . . . 364 

CHAPTER XV 

Flywheels, Pulleys and . Rotating Discs, . . . 406 

CHAPTER XVI 
Machine Frames and Attachments, ....-..,. 428 



MACHINE DESIGN 



CHAPTER I 
INTRODUCTORY 



i. The purpose of machinery Is to transform energy obtained 
directly or indirectly from natural sources into useful work for 
human needs. Useful work involves both motion and force, hence 
the basis of Machine Design is the laws that govern motion and 
force. 

The term useful work carries with it the idea of definite motion 
and definite force, for work itself is always of a definite or meas- 
urable character. An examination of any machine will show 
that its parts are so put together as to give definite constrained 
motion suitable for the work to be done. The constrainment of 
motion is determined by the moving parts, the stationary frame 
and the nature of the connections between them. 

Mechanics is the science which treats of the relative motions 
of bodies, solid, liquid, or gaseous, and of the forces acting upon 
them. 

Mechanics of Machinery is that portion of pure mechanics 
which is involved in the design, construction, and operation of 
machinery. It has been noted that the consideration of a ma- 
chine involves constrained motion, hence that portion of pure me- 
chanics is mostly needed in Machine Design which deals with 
stationary structures and constrained motion. While the laws 
of Mechanics of Machinery give us the underlying principles on 
which machine action rests, their practical application brings in 
many modifying conditions. 

Machine Design therefore may be defined as the practical ap- 
plication of Mechanics of Machinery to the design and construc- 
tion of machines. 



2 MACHINE DESIGN 

A Mechanism is a combination of material bodies so con- 
nected that motion of any member involves definite, relative, 
constrained motion of the other members. A mechanism or com- 
bination of mechanisms which is constructed not only for modify- 
ing motion but also for the transmission of definite forces and for 
the performance of useful work is called a machine. A machine 
consists of one or more mechanisms; a mechanism, however, is 
not necessarily a machine. Many mechanisms transmit no energy 
except that required to overcome their own frictional resistance, 
and are used only to modify motion as in the case of most engi- 
neering instruments, watches, models, etc. 

A brief reflection will show that the same mechanism will serve 
for different machines (see any treatise on Kinematics) and 
within limits the design of the mechanism for a given machine 
may usually be carried out, so far as motion is concerned, with lit- 
tle regard to the amount of energy to be transmitted. This, of 
course, does not apply to such mechanisms as centrifugal gover- 
nors, or in general where inertia or other kinetic actions affect 
constrainment of motion. Except for the limitations of such 
cases as those just noted, the design of any machine may be di- 
vided into two main parts : 

(i) Design of the mechanism to give the required motion. 

(2) Proportioning of the parts so that they will carry the 
necessary loads due to transmitting the energy, without undue 
distortion or practical departure from the required constrained 
motion. 

(1) The design or selection of the mechanism for a machine 
is governed by the manner in which the energy is supplied and 
the character of the work to be done; for energy may be sup- 
plied in one form of motion and the work may have to be done 
with quite a different one. If mechanisms already exist which 
will accomplish the desired result the problem is one of selection 
and arrangement of parts. But if a new type of machine is to 
be built, or a new mechanism is desired, the solution of the mo- 
tion problem borders on or may indeed be of the nature of inven- 
tion. While it is true that in most cases the mechanism and the 
relative proportions of its parts can be designed to suit the work 



INTRODUCTION 3 

to be done without reference to the energy transmitted, in general 
it is necessary to know something about the energy trans- 
mitted before any definite dimensions of the parts of the mechan- 
ism can be fixed, and frequently before the nature of the mech- 
anism is determined. Furthermore, the methods and available 
facilities of construction control the design to a large extent. 
Thus in designing a steam engine the size of the cylinder must 
be first fixed before the length of crank and connecting-rod can 
be fixed, and in general while the mechanism can be treated apart 
from the energy problem it is necessary to keep the latter con- 
stantly in mind. 

(2) The problem of proportioning the various parts of a ma- 
chine so that they will carry their loads without excessive or un- 
due deformation may conveniently be divided into two parts: 

(a) Solution as a whole, of the energy and force problem in 
the mechanism. 

(b) Assigning of dimensions to the various parts based on 
the forces acting upon them. 

(a) When the type and proportions of the mechanism have 
been fixed the relative velocity of any point in the mechanism 
may be found. If then the energy which the mechanism must 
transmit is known, it is possible, in general, to find the forces act- 
ing at any point since the law of Conservation of Energy under- 
lies all machines; or the product of velocity multiplied by force 
is constant throughout the train. If the forces acting on a 
machine member and the manner in which it is connected are 
known, these may serve as a basis for the assigning of definite 
dimensions to the part. A fuller discussion of this important 
principle is given in Chapter III. 

(b) If the forces acting on a machine member can be deter- 
mined it would seem easy to choose the material and assign pro- 
portions to it based on the laws of Mechanics, and such is the 
case when the stresses are simple and the conditions fully known. 
Thus a machine member subjected to simple tension within 
known limits, can be intelligently proportioned in this manner. 
But in many cases the forces acting are very complex, the theo- 
retical design is not always clear, and our knowledge of materials 



4 MACHINE DESIGN 

and their laws is limited in many respects. Recourse must there- 
fore often be made to judgment or to empirical data, the result of 
experience. . Even when the conditions are clear, theoretical de- 
sign must always be tempered with practical modification and by 
constructive considerations, etc. The logical method of propor- 
tioning machine elements where theory is applicable is, therefore, 
as follows : 

(a) Make as close an analysis as possible of all forces acting 
and proportion parts according to theoretical principles. 

(b) Modify such design by judgment and a consideration 
of the practical production of the part. 

In the case of details and unimportant parts, judgment and em- 
pirical data are commonly the best guides. 

Summing up then, the logical steps in the design of a machine 
are as follows: 

(I) Selection of the mechanism. 
(II) Solution of the energy and force problem. 

(III) Design of the various machine members so they will 
not unduly distort or break under the loads carried. 

(IV) Specification and Drawing. 

The last step, Specification and Drawing, is a necessary and 
important adjunct to the process of design; it is a powerful aid 
to the designer's mental process and is the best way of showing 
the workman what is to be done to construct the machine in ques- 
tion, and also of making a record of what has actually been done. 
It is not machine design of itself, however, as machines may be 
designed and built without any drawings. It is, nevertheless, 
an indispensable part of the designer's equipment. Very often 
written specifications accompanying the drawings are not only 
useful but necessary. In fact the highest skill on the part of the 
designer is often needed to clearly and fully specify in writing 
just what is to be done, as the writing of specifications presup- 
poses the most intimate knowledge of theory of design, and selec- 
tion of materials. 

From the foregoing it is seen that the part of Machine Design 
included in Mechanism can be and generally is for convenience 



INTRODUCTION 5 

taught as a separate subject, and the student is expected to have 
a knowledge of Mechanism, Mechanical Drawing, Mechanics of 
Engineering, and Materials of Engineering as a preparation for 
the work contained in this book. The chapters that follow deal 
therefore with the solution of the Energy and Force Problem, 
and the Design of Machine Elements. 



CHAPTER II 
THE ENERGY AND FORCE PROBLEM 

2. From the law of Conservation of Energy it is known that 
energy can be transformed or dissipated but not destroyed. 
Therefore all the energy supplied to any machine must be ex- 
pended as either useful or lost work. Since frictional resist- 
ances, and frequently other losses, occur in all machines, the 
useful work done must always be less than the energy received. 
The useful work delivered divided by the energy received is 
called the efficiency of the machine. This expression is differ- 
ent for different machines and is evidently a fraction or less than 
unity. In the discussion which follows in this chapter, frictional 
losses are neglected, unless otherwise stated. 

A Kinematic Cycle is made by a machine when its moving 
parts start from any given set of simultaneous positions, pass 
through all positions possible for them to occupy, and ultimately 
return to their original positions. 

The energy received by a machine during a kinematic cycle 
may or may not be equal to the work done plus frictional losses. 
Thus the energy supplied during a number of cycles may be 
stored in some heavy moving part and then be given out during 
some succeeding kinematic cycle, as in the case of a punching 
machine with a heavy flywheel. 

An Energy Cycle is made by a machine when its moving 
parts start from any given set of simultaneous energy conditions, 
pass through a series of energy changes, and ultimately return 
to their original energy conditions. 

Thus the complete mechanism of a four-stroke gas engine 
makes one kinematic cycle every two revolutions of the crank 
shaft. The slider-crank mechanism of the engine considered 
separately makes a complete kinematic cycle every revolution of 
the crank. The engine makes one energy cycle every two revo- 

6 



THE ENERGY AND FORCE PROBLEM 7 

lutions of the crank. If a punching machine driven by a belt 
and running continuously, punches a hole every fourth stroke 
of the punch, it will be making a complete kinematic cycle 
every stroke and a complete energy cycle every four strokes. 
Therefore, during a kinematic cycle, 

Energy received = useful work + lost work ± stored energy. 

And during an energy cycle, 

Energy received = useful work + lost work. 

Generally speaking, the useful work to be done and also the 
character of the source of energy are known and the problem 
of design is, therefore, to select the mechanism which will trans- 
form the motion of the source of energy into the required motion, 
to determine the capacity of the driving device, and to proportion 
the machine members. 

The proportions of any machine part depend, as regards 
strength and rigidity, on the maximum force it must carry; 
and this maximum force may be due to the direct action of the 
driving device, or it may result from the inertia effect of some 
member which has a capacity for storing energy, and in such a case 
may be greatly in excess of any direct force that the driving de- 
vice may deliver. Before this maximum force can be determined 
for any member it is therefore necessary to make a complete 
solution of the energy problem including the determination of 
the driving device. 

A knowledge of the quantity of energy required to do the 
desired work during a complete energy cycle is not always suf- 
ficient information upon which to base the design of the machine 
or the capacity of its driving device. 

A machine may receive energy at either a uniform or variable 
rate and may be called upon to do work at either a uniform or 
variable rate. Power or rate of doing work being the product 
obtained by multiplying together simultaneous values of velocity 
and force, it follows that in making any energy transformations 
both the force and the velocity factors must be kept in mind. 
While the mechanism chosen may transform the motion of the 
source of energy into the desired motion, it may not necessarily 



8 MACHINE DESIGN 

so modify the energy as to give a distribution of force at the 
point where work is being done which exactly or even approxi- 
mately fulfils the required conditions. Again, some of the 
moving machine parts may have to be very heavy in order to 
carry the required loads, and during one part of the cycle they 
may absorb energy, thus reducing the operating force, while at 
another part of the cycle they may give up energy, thus increas- 
ing the operating force. Such a condition may make an entirely 
different distribution of the forces acting on the members of the 
mechanism, from that which would occur were the parts light 
or the motion of the machine very slow, and may materially 
modify the design. 

If it is predetermined that some device is to be used for 
storing energy when the effort is in excess, and for giving it 
out when the effort is deficient, the capacity of the driving de- 
vice need only be such as will supply during the energy cycle 
an amount of energy equal to the useful work and lost work dur- 
ing that cycle. But in many machines such devices are not 
desirable and in many others they cannot be applied. 

Two such cases may be noted, (a) In many machines under 
continuous operation, where flywheels are not desirable, it is 
found that if the driving device is proportioned so as to supply 
energy at a uniform rate equal to the average rate required 
throughout the energy cycle, the force at the operating point is 
sometimes greater and sometimes less than that required. If 
simultaneous values of the force and velocity at the working 
point are multiplied together, their product is the rate at which 
work will be done at the point considered. The maximum 
product thus obtained will be the maximum rate at which work 
will be done and also at which energy must be supplied by the 
source. It is evident that the capacity of the driving device 
will be greater in such a case than if based on the average rate 
of energy required per energy cycle. If the driving device 
under the above conditions should be too large or expensive, as 
is liable to be the case in large work, recourse must be had to a 
different mechanism or to the use of flywheels or other means 
of storing and redistributing energy, (b) Again, consider any 



THE ENERGY AND FORCE PROBLEM 9 

hoisting mechanism. Not only must the driving device supply 
during the cycle of operations (the raising of the load) energy 
equal to the work done, but it must also be abje to start and 
sustain the load at any point. It is evident that in such cases 
the torque of the driving device on the hoisting drum, must be 
at least equal to that of the load, and if the torque of the driving 
device should be variable, its minimum torque must be equal 
to that of the load when referred to the same shaft.* If this 
minimum torque should be small compared to the maximum, 
the driving device chosen might have to be excessively large 
and this condition might preclude the use of the driving device 
first selected. 

In any of these cases, after the form and capacity of the 
driving device have been determined, the maximum force that 
may come on any member may also be determined. 

It is to be noted that the choice of mechanism and the capacity 
of the driving device are governed largely by the relative manner 
in which energy is to be received and work done, and it may be 
well to enumerate the combinations that can occur, before apply- 
ing the above principles to the discussion of illustrative problems. 

In any machine under continuous operation energy may be 
received and work may be done in one of the following ways: 

(a) Energy may be received at a constant rate and work be 
done at a constant rate. 

(b) Energy may be received at a constant rate and work be 
done at a variable rate. 

(c) Energy may be received at a variable rate and work be 
done at a constant rate. 

(d) Energy may be received at a variable rate and work be 
done at a variable rate. 

3. Case (a). As an example of this case, where energy is 
received at a constant rate and work done at a constant rate, 
consider a steam turbine running a centrifugal pump raising 
water to a fixed level. Evidently the rate at which energy is 

* In certain hoisting devices friction is utilized to sustain the load or prevent 
overhauling; this statement does not apply broadly to such cases. 



IO MACHINE DESIGN 

supplied must just equal the rate at which work is done plus frio 
tional and other losses, for any given period, and the capacity of 
the turbine is very easily determined. 

4. Case (b). As an example of this case (energy received at 
a constant rate and work done at a variable rate) consider the 
case of a machine for punching holes in boiler plate. Here the 
driving belt can supply energy at a constant rate while the 
useful work, which is of considerable magnitude, is delivered in- 
termittently. If the driving belt were designed with sufficient 
capacity to force the punch through the plate by direct pull it 
would have to be very large. The machine runs idly a large 
portion of the time, while the plate is being shifted, and in a 
machine of this kind a device for storing energy, such as a fly- 
wheel, can be used to advantage. The total capacity of the 
driving belt need only be sufficient to supply, during the energy 
cycle, an amount of energy equal to the useful work plus the 
lost work. When a hole is punched the velocity of the wheel is 
reduced, the wheel giving up stored energy. During the time that 
the machine is running idly the belt can store up energy in the 
flywheel by bringing its velocity up to normal. The maximum 
force that may be transmitted by the machine members will be 
based on the maximum force at the tool and will be transmitted 
only by the members that lie between the tool and the flywheel. 

As a second example of these conditions, take the design of a 
small shaping machine. Here the useful work is done during the 
forward stroke of the ram. During the return stroke frictional 
resistances only are to be overcome. The resistance of the cut 
during the forward stroke is uniform and the speed of cutting is 
limited by the character of the metal to be cut. During the re- 
turn stroke, however, the velocity may be greatly increased, the 
limiting velocity depending on the mass of the moving parts, as 
these should be brought to rest at the end of the stroke without 
shock. The machine is driven by a belt which can supply energy 
at a uniform rate and, as noted above, the work is done at a 
variable rate. 

Numerous mechanisms have been devised to meet these condi- 
tions. Suppose a mechanism such as shown in Fig. 1 has been 



THE ENERGY AND FORCE PROBLEM 



II 



selected. The maximum length of the stroke is fixed by the 
work to be done and the minimum length of stroke should be 3 
or 4 inches. Continuous rotary motion is imparted to the crank 
a through the gear b of which it forms a part. The gear b is in 
turn driven by the pinion c which is rigidly attached to the shaft 
d. On the other end of d is a stepped pulley having diameters 



Diagram 




Fig. 



D t D 2 D 3 Z> 4 . On the countershaft overhead is a mating stepped 
pulley so placed that when the belt is on the largest step of the 
machine D x it is also on the smallest step of the countershaft 
pulley. The crank pin on a is adjustable and can be moved from 
the outer position as shown toward the centre of the crank, so 
that the vibrator e can be made to give the ram R any length of 



12 MACHINE DESIGN 

stroke from the maximum (20 inches in this example) to a 
minimum of 3 or 4 inches. The range of velocity of the tool 
for any length of stroke must be such that it can be lowered to 
the cutting velocity of hard cast iron or tool steel and raised to 
the economical cutting velocity of brass. With the pin in its 
extreme outer position and the belt on the large step D x the 
speed of the ram will be a maximum for that position of the 
belt. As the. crank is drawn toward the centre (the belt re- 
maining in its original position) the velocity of the ram is obvi- 
ously decreased. If now the belt is shifted to a smaller step as 
D 2 the velocity of the ram will be increased, so that at any stroke 
variable speed may be obtained to suit the metal to be cut. It is 
not desirable to use a flywheel, the inertia of the moving parts is 
small, and the problem is therefore to design the driving belt and 
proportion the machine members on the basis of the maximum 
pull which the belt may be able to exert. 

The mechanism transforms the uniform rotary motion of the 
line shaft into the required reciprocating motion. Consider the 
crank pin at its extreme outward position and the belt on D v 
The velocity diagram for full forward stroke under these condi- 
tions is shown, the ordinates of the diagram* representing the 
velocity of the ram to the scale that the crank length represents 
the uniform velocity of the crank pin. The diagram for the 
backward stroke is not drawn since it is not needed in the 
solution of the energy problem; but it should in general be 
drawn to make sure that the change in velocity at the extreme 
ends of the stroke is not excessive. If the belt supplies energy 
at a constant rate the force which it can deliver at the tool will 
vary inversely as its cutting velocity. The cutting resistance, 
however, is uniform so that while the mechanism produces the 
desired transformation in motion it may not give the distribution 
of force desired. 

To design the driving device (or belt) for such a mechanism 

* For a full discussion of these so called quick-return mechanisms and the methods 
of drawing velocity diagrams see " Kinematics of Machinery " by John H. Barr, 
" Machine Design " by Smith and Marx and " Machine Design," Part I., by F. R. 
Jones. 



THE ENERGY AND FORCE PROBLEM 1 3 

the operating conditions of the machine when the belt has both 
its maximum and minimum velocity must be investigated. The 
maximum pull which a belt can give is 7\ — T 2 where T x is the 
allowable tension on the tight side of the belt. (See Church's 
" Mechanics," page 182.) The power* that a belt can give out is 
therefore V (T x — T 2 ) where V is the velocity of the belt. Since 
T x — T 2 has, at all moderate belt speeds, a constant maximum 
value for a given belt, the power that a belt can deliver will vary 
directly with its velocity. The belt receives its energy from a 
shaft running at constant speed and when the belt is on the 
smallest step of the countershaft cone it will also be on the 
largest step D x of the machine cone and will in consequence be 
running at its lowest velocity, under which condition its capacity 
for delivering energy is a minimum. 

The maximum power required for small machine tools is 
approximately constant at all speeds; for since the heating effect 
which governs the cutting capacity of the tool is proportional to 
the work done, it follows that as the cutting speed is increased 
the resistance of the cut must be decreased and vice versa, thus 
keeping their product approximately constant. If then the belt is 
designed to have sufficient capacity when the ram is making full 
stroke and the belt is on D 1} and hence at the lowest belt veloci- 
ty, it will have excess capacity when in any other position. If a 
softer metal is to be cut the velocity of the ram may be increased, 
but this can only be done by shifting the belt to a position where 
its velocity and hence its capacity will be greater. 

As ( before noted, the effect of moving the crank pin inward, the 
belt remaining in the same position, is to decrease the average 
velocity of the ram. Therefore as the stroke is made shorter 
the velocity of the crank, to maintain a given cutting speed, must 
be increased by shifting the belt to a smaller step of the machine 
cone. The other limiting condition is when the ram is making 
its shortest stroke and giving a cutting velocity high enough for 
the softest metal to be worked. The belt should then be on the 
smallest diameter Z> 4 , and hence at its highest speed. 

* A full discussion of the power transmitted by belting is given in chap. 12. 



14 MACHINE DESIGN 

An inspection of the velocity diagram when the ram is making 
full stroke shows that its velocity is a maximum when the ram 
is in mid position. Neglecting friction and inertia, which here 
are small, the force exerted on the ram will be a minimum where 
the velocity of the ram is a maximum at any given belt velocity, 
because, for a given belt pull since no flywheel is used, force at belt 
X velocity of belt = force at tool X velocity of tool. If, there- 
fore, with the ram making full stroke, the capacity of the belt 
when running on D x is made great enough to give a force at mid 
position of the ram equal to the required cutting force, it will 
have excess capacity at any other position; and if this condition 
does not give too large a belt the driving device will be satisfactory. 
The maximum force that any member may have to sustain will 
be based on the maximum torque of the belt, which will occur 
when it is running on D x \ for since the inertia forces are small 
this torque will be transmitted directly to the members, and the 
resulting stresses may be easily computed. 

Example : 

Let the greatest resistance of cut 

" " maximum stroke of ram 

*' " minimum stroke of ram 

" " maximum length of crank = 

" " minimum " " " 

" " max. cutting speed on shortest stroke and highest 

belt speed = 60 ft. per min. 
" " max. cutting speed on full stroke and lowest belt 

speed = 25 ft. per min. 
Then in general, 

linear velocity of crank max. linear velocity of ram 



800 


lbs. 


20 


inches. 


4 


inches. 


6^ 


(t 


i l A 


11 



length of crank max. ordinate of diagram* 

Hence in this example when the ram is making full stroke at 
lowest speed, 

* In the mechanism here chosen the position of the ram for maximum velocity 
can be located by inspection and the value of the velocity determined without 
drawing the complete diagram. In general, however, the diagram must be drawr 
in order to locate the maximum ordinate. 



THE ENERGY AND FORCE PROBLEM 15 

25' X 6J 
Linear vel. of crank = = 23.5 ft. per min. 

. '. R.P.M. of crank = * 3 ' S X " = 6.0. 

2 X 7T X 6J 

In a similar way when the ram is making the shortest stroke 
at highest speed, 

Linear velocity of crank = 42.5 ft. per min. 

Therefore, R.P.M. of crank = -^^ ^- = 54.1. 

2 x * x ii 

Let the gear ratio be 8 to 1. Then the minimum and maximum 

R.P.M. of shaft ^ = 55.2 and 432.8 respectively. A 14" pulley 

is a convenient diameter for D v 

1 • e i 1 1 1 14 X 7T X 55.2 

. ' . velocity of Delt on low speed = £ ^— 

12 

= 204 ft. per min. 

If the efficiency of the machine be 85 per cent, the maximum 

rate of doing work at this position of belt is the cutting resistance 

multiplied by the maximum velocity of the ram, divided by the 

800 
efficiency, or — — X 25 = 23,500 ft. lbs. per minute. 
• 05 

22 coo 
.*. effective pull at belt = — - — =115 lbs. approximately. 

The effective pull of single-ply belt per inch of width may be 
taken at 40 to 45 lbs. 

.*. width of belt = — = 2^" nearly. 

45 

If the cone pulleys on machine and countershaft are alike, as is 
the usual case in metal-working tools, then 



Max. R.P.M. of Machine Cone 



£1 iMz 

D 4 ^Min. R.P.M. of Machine Cone' 



R.P.M. of Machine Cone 



Max. R.P.M. of Machine Cone ' 

and hence, in the example if D x = 14, Z> 4 = 14 -J ^' 2 = 5' 

\ 432.8 
nearly. 



1 6 MACHINE DESIGN 

The maximum force that may be applied to any member 
will be based on the maximum torque of the driving belt, which 
occurs when the belt is on D x the largest step of the machine 
cone. The difference in this respect between this case and the 
punching machine discussed above should be noted, for, while 
the driving mechanisms of both can deliver energy at a uniform 
rate and while both do work at a variable rate, the maximum 
load is applied in entirely different ways. 

During the complete energy* cycle of the machine the total 
work done, neglecting friction, is equal to the length of stroke 
multiplied by the uniform resistance of the cut, or 

20 
800 X — = 1333 ft. lbs. For every cycle of the machine the 

shaft d makes 8 revolutions; hence the amount of energy that the 
belt could deliver if work were done uniformly during one cycle 

. 14 X 7T 

is 8 X — X 115 = 3370 ft. lbs. 

The capacity of the belt is therefore two and one-half times as 
great as it would need to be if a device for equalizing the energy, 
such as a flywheel, had been used. Where a small machine is 
belt-driven, as in the case discussed, this added first cost is not 
serious. But when the power needed is great, or in such cases as 
direct driving by electric motor, the additional cost of a driving 
device so greatly in excess of average requirements needs to be 
carefully considered. This, in fact, is one of the most important 
elements to be considered in fixing the size of motors needed 
for direct-driven machine tools, sometimes making it desirable to 
introduce a flywheel to reduce the size of motor. 

5. Case (c). One of the best examples of Case (c) where 
energy is received at a variable rate and work is performed at a 
uniform rate is found in the reciprocating steam engine, and 
since this machine is of such great importance to the engineer 
it will be discussed somewhat in detail. Here the energy is 
supplied in the form of steam pressure, and after cutoff occurs 
and the steam expands in the cylinder the pressure falls from 

*The kinematic and energy cycle are, in this case, simultaneous. 



THE ENERGY AND FORCE PROBLEM 



17 



the "initial" or boiler pressure to somewhat above exhaust or 
atmospheric pressure. The energy is therefore supplied at a 
varying rate. But the engine is required to deliver energy at 
the driving belt at a uniform rate. The mechanism used will 




— Stroke 



Fig. 3. 



1 8 MACHINE DESIGN 

produce the required transformation of the reciprocating motion 
of the piston into the rotary motion of the crank shaft. But the 
distribution of the driving force in the form of torque or tan- 
gential effort will not be uniform but it will be a maximum 
somewhere near the position at which the crank is at right angles 
to the connecting-rod, and it becomes zero when the crank is on 
the dead centre. The turning effort will therefore sometimes 
be greater and sometimes less than the resisting effort of the 
driving belt and the machine will stop unless a redistributing 
device, such as a flywheel, is used. The reciprocating parts, 
such as the piston and crosshead, and also the connecting-rod, 
are heavy and their maximum velocity is considerable; hence 
the forces due to their inertia cannot be neglected. 

Referring to Fig. 2 (a) , the crank a is required to rotate around 
the center O with uniform velocity and to give a uniform force 
at the driving belt. The moment at the driving belt is equal to 
the average moment at the crank pin, hence the equivalent 
uniform force at the crank pin may be derived from that at the 
belt. This required driving force at the crank pin may be 
plotted radially from the crank circle as a base, forming a polar 
diagram of the required force at the pin, as shown by circle S. 
The crosshead C moves at a varying rate of speed. If the 
velocity of the crank pin be represented by the length of the 
crank, the intercept Op made by the connecting-rod on the ver- 
tical through O will represent the simultaneous velocity of 
the crosshead to the same scale. These intercepts may be plot- 
ted at the corresponding positions of the crosshead, thus out- 
lining the curve whose ordinates represent the velocity of the 
crosshead at any point. 

The forces acting upon the piston and which must be trans- 
mitted to the crank are, 

(1) The steam pressure which is represented at any point by 
the ordinates of the curve T, Fig. 2 (b). 

(2) The back pressure* on the other side of the piston, act- 

* This generally amounts to 2 or 3 pounds per sq. in. above atmospheric pressure 
in non-condensing engines. 



THE ENERGY AND FORCE PROBLEM ig 

ing against the steam pressure, and represented by the exhaust 
pressure line z z and the compression curve U. 

(3) The inertia forces due to accelerating and retarding the 
heavy reciprocating parts. 

During the first part of the stroke these inertia forces tend to 
reduce the effective pressure transmitted to the crank pin, and 
during the latter part they increase the effective force on the rod. 
They can be represented graphically by such a curve as V, The 
first two curves can be found by the well-known methods of 
drawing indicator cards, and the third can be found either by 
mathematical deduction or by graphic methods* based on the 
velocity diagram. It is believed that the analytical method is 
the most satisfactory, and such a method is presented in a suc- 
ceeding article. 

If the acceleration is known the force necessary to produce the 
acceleration is also known since accelerating force = mass X ac- 
celeration, and the force at any point (reduced to pounds per 
sq. in. of piston) may be plotted as shown by curve V, Fig. 2 (b). 
When the reciprocating parts reach their maximum velocity 
their acceleration is zero, hence the curve of acceleration forces 
crosses the axis at a point g corresponding to the point of maxi- 
mum velocity. This point is very nearly at the position where 
the crank and the connecting-rod are at right angles and the 
error introduced by assuming this to be so is small with ordinary 
ratios of crank to connecting-rod length. Beyond g the recipro- 
cating parts are retarded, hence the inertia forces increase the 
effective crank-pin pressure from that point on. The compression 
curve (U) tends to decrease the effective pressure on the piston 
and hence its ordinates must be subtracted from the forward 
pressure. The algebraic sum of the curves T, U, and V will give 
a resultant pressure curve W, Fig. 2 (c) , whose ordinates at any 
point represent the effective pressure acting on the piston rod at 
that point. This effective pressure is transmitted to the crank 
by the connecting-rod b. The pressure of the rod against the 
crank pin may be resolved into two components, one tangential 

* For a full discussion of this matter see " Kinematics of Machinery" by J. H. 
Barr, page 71, paragraph 42. 



20 MACHINE DESIGN 

to the crank circle and tending to produce rotative motion, and 
one radial along the crank tending to produce compression or 
tension in the crank and friction in the main bearing. Only the 
tangential force can do useful work. If friction be neglected 
the rate at which work is done by this force at the crank must 
equal the rate at which work is being done at the piston. Now 
the curves R and W, Fig. 2 (a) and 2 (c) respectively, give the simul- 
taneous values of force and velocity at every point of the stroke. 
If such simultaneous values be multiplied together and divided 
by the uniform velocity of the crank (all in the proper units) 
the quotient is the tangential force at the pin, and this may be 
plotted radially on the crank circle as a base, thus giving what is 
called a radial crank-effort diagram, Fig. 2 (c), Curve X. 

These values of the tangential force can be found more easily 
graphically. It will be remembered that the ordinates of the 
velocity diagram (R), as drawn in Fig. 2 (a), represent the velo- 
city of the crosshead to the same scale as the length of the crank 
represents the velocity of the crank pin. In Fig. 2 (c), the connect- 
ing-rod extended, if necessary, cuts the perpendicular through 
O in the point h. Therefore O h = velocity of crosshead when 
O j = velocity of crank pin. Neglecting friction, the rate of 
work at the crank pin is equal to the rate of work at the cross- 
head, hence the velocity of the crank pin multiplied by the force 
at the crank pin is equal to the velocity of the crosshead multi- 
plied by the force at the crosshead, or the tangential force xO j 
=eJ x XOh. 

. \ tangential force = 1 * — 

Ok Oi 

Lay off O i = e^ and draw i k parallel to b. Then, jr-r = —. 

m e ^1 OixOh e x j x xOh . _ , 

Therefore, O k = ^— — = ^—. = tangential force. 

O 1 O j 

Therefore O k may be laid off radially from ; as an ordinate of 
the required curve as j k'. The construction for the return 
stroke is performed in a similar manner. 

It will be noted that the distribution of force as represented 
by this diagram is less uniform than the original curve of press- 



THE ENERGY AND FORCE PROBLEM 21 

ure at the crosshead. By the conditions of the problem, how- 
ever, the mechanism must produce a uniform turning effort at 
the driving belt or such as would be given by a crank-effort dia- 
gram like S, Fig. 2 (a) . A flywheel must therefore be used to 
store energy when the crank effort is in excess and to give out 
energy when the crank effort is deficient. Fig. 2 (d) shows the 
crank-effort diagram rectified with rectangular ordinates equal 
to the polar ordinates of curve X. The base YY is equal to the 
circumference of the crank circle and the ordinates of the line 
I m are equal to the ordinates of the required uniform crank- 
effort curve S. Since the abscissas represent space and the ordi- 
nates represent force, the areas /, K, J, I 19 K 1} etc., represent 
work. The work represented by K+K 1 is that which the fly- 
wheel must absorb and the area represented by / + J + I X + J x that 
which it must give up in one revolution. Manifestly I + J + I 1 + 
J t must equal K+K v A full discussion of the design of the 
flywheel will be given in a later chapter. 

The maximum force that may come upon the crosshead can 
be seen from an inspection of the force diagram W. It is to be 
noted in this regard that if the engine is designed for variable 
cutoff, an indicator diagram at late cutoff should be drawn for 
the purpose of locating this maximum force, as an earlier cutoff 
will not give the maximum value. The method of analysis de- 
veloped above will enable the designer to determine the maxi- 
mum straining action on any member of the mechanism. 

The graphical method of finding the inertia curves, while con- 
venient, are open to criticism on account of their inaccuracy be- 
cause the tangents or sub-normals to the curve, on which these 
graphic methods depend, are difficult to construct with accu- 
racy and are at some points indeterminate. In general, there- 
fore, it is thought that the following method or some similar 
one is more satisfactory. 

Referring to Fig. 3 (page 17), 

Let a = acceleration at any point, 
" R = length of crank in feet, 
" L = " " connecting-rod in feet, 



22 MACHINE DESIGN 

Let N = Rev. per min., 

" and <p = angles made with centre line by the crank and 
connecting-rod respectively at any position measured from the 
crank position O r, 

Let k = distance from centre of crank shaft to mid position of 
crosshead, 

Let x = displacement of crosshead from mid position, 

71 ~ R' 
" v = velocity of crosshead at any point x, 
" t = time elapsed corresponding to v. 
" to = angular velocity in radians per second, 

Then x + k = O B + B C = R. cos e + L cos <p, 
But L cos <p = VL 2 —R 2 sm 2 d = R J— 2 — sin 2 
= RV n 2 — sin 2 



. *. x + k = R (cos + Vn 2 — sin 2 0) (i) 

Expanding the radical by the binomial theorem and omitting all 
terms beyond the second (which can be done without appreciable 
error with the limiting proportions ordinarily used) equation (i) 
becomes, 

x + k = R\ cos e + (n — J | (2) 

Now x = the distance moved through by the crosshead, from mid 

d x 
stroke and velocity at x = — ; and therefore differentiating 



(2) with reference to t 


dx 

v — -r- 

dt 


= — R 


The acceleration 


= 


dv 
a = — - = 
dt 


d 2 x 
' dt 2 " 



(sin 



sin 2 0\do 

+ -^r)Tt ■ • • • & 



„ / C0S2#\ /^0\ 2 

-R[coso + -—)[ T J . . (4) 



COS2#\ /d0\ 2 
^n~) \Tt> 

but t~ = angular velocity in radians per sec. = — — , 
dt n J 60 

/2- AT\2 / COS2 0X 

hencea = -(-— Jtf^cos0 + -^J, (5) 



THE ENERGY AND FORCE PROBLEM 23 

which is the general expression for acceleration of the recipro- 
cating parts. 

If the weights of parts be called W, from Mechanics it is 
known that the force necessary to produce an acceleration (a) is 

W 

P = — a where g = 32.2 in English units; therefore 

W-f2*N\ 2 ( COS20\ , _ . . , 

P = — — R[—7 — j I cos + where R is in ft. . (6) 

g V 00 ' v n ' . 

or reducing, 

WrN 2 / cos 2 e 



P = 



(cos e + ) where r is in inches. . (7) 



expression — R I— — J is the centrifugal force of a weight 



When the solution of the above expression gives a negative 
result the force of inertia is acting away from the crank and 
when positive, toward the crank. It is also to be noted that the 
W f2*N\ 2 . 

g V 60 

equal to that of the reciprocating parts concentrated at the crank 

. . . . . , WR"> 2 
pin since centrifugal force in general is equal to . 

By means of equation (7) all points on the acceleration curve 
could be found and plotted. In general, however, the exact 
characteristics of the curve are not essential and it is sufficient to 
make the three most simple solutions as follows, and a curve 
drawn through the three points thus located is sufficiently accu- 
rate for all ordinary purposes. In cases of extremely high speed 
with small ratios of connecting-rod to crank a more accurate de- 
termination of the curve may be desired. 

When = 0, P = — (Ell?) L + I) . m ( 8 ) 

\ 35200 / \ n / 

When = 180 , P = + (^1^1) (1 --) . . (9) 

\ 35200 / >• n' 

(W rN 2 \ /i\ 

When = oo° or 270 * P = ( ) ( -) . . (10) 

' V. 35200 / ^n* 

* The piston is not at half stroke- 



24 MACHINE DESIGN 

If the inertia forces are to be combined with the steam press- 
ures, as shown graphically in Fig. 2 (b) , they must be reduced to 
pounds per square inch of piston to give correct diagrams. 

An example may serve to make these points clearer. Let it 
be required to design a steam engine to deliver 150 H.P. with 
the following data: 

Steam pressure = 90 lbs. gauge. Cutoff at y% stroke. 

Ratio of crank to connecting-rod = 1 to 5. 

Piston speed = strokes per minute multiplied by length of 
stroke = 640 ft. 

Here something must be known about the size of cylinder 
necessary, before definite dimensions are assigned to the various 
members. Let a theoretical indicator card be drawn as in Fig. 
2 (b), neglecting for the present the inertia curve V since this 
only tends to redistribute the energy and does not affect its 
quantity. The distance z z represents the piston travel and the 
ordinates of the curve T represent piston pressures; therefore 
the area between z z and the curve T represents the work done 
by the steam pressure during the stroke. In a similar way the 
area under curve U represents the work of compression due to 
back pressure. The difference of these areas is the net work 
done per stroke of piston and the mean ordinate corresponding 
to this area represents to the proper scale the average pressure 
per sq. inch on the piston during stroke. In the case given 
z z = 2". Area under T minus area under Z7 = 1.75 sq. in. There- 

1 .75 
fore mean ordinate = = .875". The scale of pressures 

taken is 1" = 70 lbs. Therefore mean pressure during stroke 
= 70 X .875 = 62 lbs. 

Let A = area of piston. 

P = mean effective pressure per sq. in. 
L = length of stroke in feet. 
N = number of revolutions per minute. 
H.P. = horse power required. 

2 P LA N 

Then H.P. = . Here P.N X L and H.P. are known. 

33000 



THE ENERGY AND FORCE PROBLEM 25 

77. P. X 33000 150 X 33000 

Whence A = — =— — Ar , = —, — -~7 =132 square inches, 

P X 2 N L 62 X 640 ^ 

or a diameter of cylinder of 13 inches. 

If the stroke be taken at about twice the diameter of the cyl- 
inder, or say 24 inches, the proportions will be good. 

Hence since 2L X AT = 640, N = i6o R.P.M. The mechan- 
ism can now be laid out to scale. This has been done in Fig. 2 
(a and c),* the space scale being i" = i ft. 

As before stated, the location of the three points, namely, 

where is respectively o°, 180 , and 90 or 270 (Fig. 3), is 

sufficient to locate the inertia curve. In the above example 

W = 3.5, n — 5, and N = 160. 

The general expression for the inertia force is, for = 0. 

WrN 2 ( 1 \ / 1 \ 

P = (1 + — ) = C I 1 + — ) where C is a constant 

35,200 ^ n' v n' 

3.5 X 12 X 160 2 

and here equal to == 30. ?. 

4 35.2QO ^ 



6 lbs. 



Therefore, When = o°, P = 30.5 ( 1 + -) = 36. 
When <? = 90 , P = 30.5 (-) =6.1 lbs. 

When <? = 180 , P = 30.5 ( 1 — -) = 24.4 lbs. 

These values serve to locate the curve as in Fig. (2). 

The resultant of T U and V, curve W, Fig. 2 (c), can now be 
drawn and the crank-effort diagram X plotted. The crank-effort 
curve can be rectified as in Fig. 2 (d) and the mean ordinate Yl 
drawn. The area 7 + 7= K will be proportional to the energy 
to be absorbed and delivered by the flywheel. One inch of ordi- 
nate here = 70 lbs. per sq. in. of piston and one inch of abscissa 
= 1 f t. ; therefore one sq. in. of area = 70 ft. lbs. 

The area of 7T = .5 sq. in. and area of piston = 132 sq. in. 
Hence, if E = energy to be absorbed, 

£ = .5X70X132=4,620 ft. lbs. on which the de- 
sign of the flywheel can be based. 

* Reduced in reproduction about one-half. 



26 MACHINE DESIGN 

The maximum pressure that can occur on the piston is the 
initial or boiler pressure as the ordinates of W are at all points 
less than those of T. Hence, when running, the parts will be 
subjected to less load than in starting up, when full boiler press- 
ure may be applied before inertia forces become noticeable. 

6. Case D. A good example of energy supplied at a vary- 
ing rate and work done at a varying rate is found in a direct- 
driven air compressor. Here the varying steam pressure in the 
steam cylinder is opposed by a varying air pressure in the air 
cylinder as shown in Fig. 5 (a) . The area of the cylinders are, 
for simplicity, assumed to be equal. The steam cylinder takes 
steam at 80 lbs. pressure and the air compressor cylinder delivers 
air at 100 lbs. pressure. The efficiency of the system shown is 
taken at 80 per cent, and hence the area of the compressor card 
is 80 per cent, of the steam card.* If both the pistons were 
rigidly attached to the same rod it is evident that the maximum 
steam pressure will occur where the air pressure is a minimum. 
If, however, each cylinder is independently connected to a com- 
mon shaft by means of a crank and connecting-rod mechanism, 
the maximum and minimum pressures of the cards may be made 
to coincide more closely by placing the crank pins at the proper 
angular distance apart. In other words the mechanism may be 
so designed that energy will be delivered at the working point 
more nearly at the rate required by the work to be done. The 
loss by friction, etc., is about 20 per cent. Part of this is lost 
on the steam side and part on the air-compressor side. It can 
be assumed, without great error, that the losses can be evenly 
divided between the two slider-crank chains and also that the 
loss is at a uniform rate throughout the stroke. Thus the loss 
on the steam side can be represented by the line a b, Fig. 5 (a) , 
which reduces the effective pressure at every point by a fixed 



* In the general case, where the cylinders are of different diameter and area, 
the diagrams which represent pounds per square inch of piston area would not 
have a ratio equal to the efficiency. The mean effective pressure of the air cylinder 
multiplied by the area of the air cylinder, divided by the mean effective pressure of 
the steam cylinder multiplied by the area of the steam cylinder, would, in this case, 
equal the efficiency. 



THE ENERGY AND FORCE PROBLEM 



27 



amount. In a similar way ordinates to the line c d increase the 
effective resistance of the air diagram. The area of the diagrams 
modified in this way will be equal and all energy supplied will 
be accounted for. 




Fig. 5 (b). 




Fig. 5 (c). 




Fig. 5 (d). 



Since the moving parts of both slider-crank chains will be 
heavy, the effect of inertia cannot be neglected. In Fig. 5 (b) 
the air and steam cards are shown with the inertia curve, the 



28 MACHINE DESIGN 

friction line, and the compression curves in their correct relation- 
ship. Fig. 5 (c) shows the resultant pressure curves, the curve 
of air pressures being plotted below the base line for conveni- 
ence. The crank-effort curve of the steam cylinder is repre- 
sented by X, and the resisting crank-effort curve of the air cyl- 
inder is represented by Y. The cranks are here placed 90 
apart, the steam crank being in advance, a common arrangement 
in practice. It is evident, however, that this is not the most 
advantageous angle, for if the point e on the air curve is made 
to correspond with / on the steam curve, Fig. 5 (c) , the excess 
and deficiency of effort will be still further reduced. This would 
place the cranks at 45 apart. This is even more clearly shown 
in Fig. 5 (d), on the rectified curve of crank effort. Here the 
area K+K x is the amount of energy to be absorbed and I + J + 
Iy + Ji the amount to be given up by the flywheel during one 
revolution. In the steam slider-crank mechanism the greatest 
pressure is, as before, that due to the initial steam pressure, 
while on the air side it will be that due to the terminal air 
pressure. 

6.1. In the four cases discussed above the action of the ma- 
chine has in all instances been supposed to be continuous, and all 
machines which operate continuously will belong to one of these 
classes. Where the action of the machine is intermittent or 
irregular, these general solutions will not always hold and the 
design of the machine cannot be based on the energy given or 
•received, but will depend on the maximum force or maximum 
torque or, in other words, on the mechanical advantage which 
the motor must possess. Thus the motor on an auto car has a 
certain maximum capacity for delivering power. On a level 
road it can propel the car at a high rate of speed, the engine 
making only a few turns to every revolution of the wheels. But 
on a steep hill the gears must be shifted so that the engine has a 
greater mechanical advantage, and gives a greater torque on the 
axle, the engine making many revolutions to every one of the 
wheels. Another example of this is the case of hoisting mech- 
anisms already discussed somewhat (see article 2). An en- 
gine or a motor might be capable of giving out energy at a rate 



THE ENERGY AND FORCE PROBLEM 29 

equal to that required to lift the load in a given time, and it 
might be able, running continuously, to raise the load to the 
required height. But its ability to start and sustain the load at 
any point will depend on whether it has a mechanical advantage 
at that point and not on its capacity. Where the torque of the 
load is constantly changing, as in deep mine hoisting, the design 
of the hoisting devices becomes quite complicated and is beyond 
the scope of the present treatise. It will be noted, however, 
that in such cases the minimum torque of the motor or engine 
must always exceed the maximum torque of the load when re- 
ferred to the same shaft. This general principle must be kept 
in mind in designing hoisting devices and similar machines 
which act intermittently and slowly, or where redistributing 
devices are undesirable or impossible. 

6.2. Redistribution of Energy and Inertia Effects. Devices 
for storing and redistributing energy are very common in transmis- 



Fig. 5 (e). 

sion systems. Thus, in hydraulic distribution, the excess supply 
of power is stored in an accumulator, and given out again when 
the supply is deficient. In electrical distribution a storage battery 
is sometimes used for the same purpose. In transmission of power 
by compressed air a large reservoir is sometimes employed as a 
store-house of energy. In the case of a single machine, the re- 
distribution is effected by compressing a gas, by using a spring, or 
by accelerating and retarding some heavy moving part. Thus 
in the steam engine the piston compresses steam in the clearance 
space at the end of its stroke, and the energy so absorbed is re- 
turned to it during the next stroke. Again, when the energy 
supplied by the steam is in excess of the effort required, the fly- 
wheel absorbs the excess and thereby has its velocity (and hence 



30 MACHINE DESIGN 

its kinetic energy) increased. When the effort is in excess, the 
wheel gives up the stored energy at the expense of its velocity. 

It does not necessarily follow, however, that all heavy moving 
parts simply redistribute the absorbed energy as useful work, 
as the action may be a positive source of loss. In Fig. 5 (e) let 
A be the platen of a large planing machine, and suppose it to be 
making its return stroke, moving from left to right. The force 
just necessary to slowly move the platen may be represented by 
the vertical ordinates of the diagram abed. Suppose now, 
that a greater force is applied, in order to hasten the operation, 
so that at the position A', the platen has been accelerated till its 
kinetic energy is equal to the rectangle e g h c. Evidently the 
platen will not stop at the end of the stroke if the actuating force 
be removed at A' f as the work of friction during the remainder 
of the stroke is less than the stored energy. If, therefore, the 
"return "belt is removed at A ' and the " driving " belt applied, the 
latter will slip upon the driving pulley till the excess of energy 
is absorbed and dissipated as heat. If the point A ' has been prop- 
erly chosen the platen will just stop at the end of the stroke and 
the energy absorbed by the belt will equal the area / g h b. If 
a spring, 5, were fitted to the machine, so that the work of com^ 
pression from the position A' to the end of the stroke just equalled 
the excess kinetic energy of the platen, at that position, the return 
belt could be thrown off at A', and the platen would stop at the 
end of the stroke. The energy stored in the spring would then 
be returned to the platen on the forward stroke. This latter 
action is identical with that of compression in the steam-engine 
cylinder, Fig. 2, the energy under the curve U being returned to 
the reciprocating parts on the next stroke. It is to be noted in 
this last case, that even if the work of compression is not quite 
equal to the energy to be absorbed during the latter part of the 
stroke, there is no loss of energy (friction neglected), as what is 
not absorbed by compression is absorbed at the crank pin in 
useful effort. 



CHAPTER III 
STRAINING ACTIONS IN MACHINE ELEMENTS 

7. Nature of Forces acting in Machines. From the fore- 
going chapter it is clear that machine members which transmit 
energy are subjected to forces of a varying character and inten- 
sity. Since the various parts of a machine must be constrained 
to move in fixed paths it is important that they should neither 
break or be distorted appreciably under the loads carried; that 
is, the members must be not only strong but also stiff. The pro- 
portioning of machine elements as dictated by various methods 
of loading is therefore most important, and will be considered in 
this chapter. 

The forces acting on a machine element may be one or several 
of the following: 

(a) The useful load due to the energy transmitted. 

(b) Forces due to frictional resistances. 

(c) The weight of the part itself or of other parts c 

(d) Inertia forces due to change of velocity. 

(e) Centrifugal or inertia forces. 

(f) Forces due to change of temperature. 

"(g) Magnetic attractions, as in electrical machinery. 

These forces or loads may be applied to a machine in several 
ways. They may act steadily in one direction ; they may act in- 
termittently in one direction, or they may be applied first in one 
direction and then in the reverse ; they may be applied gradually, 
or suddenly in the nature of a shock. 

A steady or dead load is one which is always applied steadily 
in the same direction. A live load is one which is alternately 
applied and removed. A suddenly applied load is one imposed 
instantaneously but without initial velocity. If the load is ap- 

21 



32 MACHINE DESIGN 

plied with initial velocity as in the case of a blow from a falling 
body, the member is subjected to impact. 

8. Nature of Straining Actions, Stress and Strain. Since 
all materials of construction are more or less elastic a machine 
element must change its form to some extent whenever subjected 
to a load. This change of form may be very small and tempo- 
rary; it may be a permanent distortion; or if the load applied 
be heavy enough the element may even be ruptured. Such 
change of form, whether temporary or permanent, is called a 
strain. When a machine member is thus distorted under a load 
certain molecular reactions, equal and opposite to the load applied, 
are set up within the material and resist the deformation. Stress 
is the term applied to this internal reaction and is to be clearly 
distinguished from strain, stress being in the nature of a force 
and strain being a dimension. 

The character of the straining action and of the stress which 
results from a given load depend upon the direction and point of 
application of the load (or forces), and upon the form, the posi- 
tion, and the arrangement of the supports of the member. A 
given load may produce tension, compression, shearing, flexure, 
or torsion or a combination of these. Of course tension and com- 
pression cannot both exist at the same time between any pair of 
molecules. Flexure is a combination of tensile and compressive 
stresses between different sets of molecules; or, as it is often ex- 
pressed, in different fibres* of the same body. Torsion is a 
special form of shearing stress. Owing to the frequent occur- 
rence of flexure and torsion it is convenient to treat these as 
elementary forms of stress. 

The stresses due to tension, compression, and flexure are essen- 
tially molecular actions normal to the planes separating adjacent 
sets of interacting molecules; that is, the stresses increase or de- 
crease the distances between these molecules along lines connect- 
ing them. 

The primary straining effect of shearing and torsional actions 
is displacement of adjacent molecules, between which the stress 

* It should be noted that the term fibre is used in a conventional sense when 
discussing homogeneous metals, such as iron and steel. 



STRAINING ACTIONS IN MACHINE ELEMENTS 2)3 

acts, tangentially to the planes separating such molecules. In 
uniform shear the interacting molecules move or are strained rel- 
atively with a rectilinear translation. In torsional action the ad- 
jacent molecules each side of a plane of stress have a relative 
motion or strain about an axis. A brief reflection will show that 
in reality only two kinds of strain exist, namely, elongation (con- 
traction if negative) and shearing. In a similar way only two 
corresponding kinds of stress are met with, namely, normal or 
direct, and tangential or shearing. But for convenience it is 
much more desirable to treat the special cases previously men- 
tioned, separately as elementary stresses. (See Church's 
Mechanics, page 201.) 

Machine members are often subjected to combinations of these 
simple stresses, as flexure and torsion. Such stresses are called 
Compound Stresses and will be more fully treated later. 

When a load is applied to a piece of material the strain which 
results is a function of the load and of the character of the ma- 
terial involved. In general for a given loading the deformation 
is different for different materials but constant in its relation to 
stress for any one material. These relations have been deter- 
mined experimentally for all the ordinary materials used in engi- 
neering, and works on mechanics of materials treat of the sub- 
ject fully. Enough will be inserted here to make the discussion 
complete. 

If a bar of metal is tested under an increasing tensile load and 
the strain caused by each successive load is accurately observed the 
relation between stress and strain can be shown graphically as at 
O a d e Fig. 6; such a diagram is called a stress-strain diagram. 

If axes O X and O Y are chosen and the stresses plotted as 
ordinates and strains as abscissas, it will be found that up to a 
certain point as a, either in tension or compression, the curve so 
formed is sensibly a straight line; that is, stress is proportional 
to strain. Further, if at any point below a the stress is released, 
the piece returns to its original shape. But above a this relation 
ceases, strain usually increasing* faster than stress, till finally 

* Ordinary rubber is an exception to this general rule, strain decreasing as 
stress increases. 
3 



34 



MACHINE DESIGN 



rupture occurs. If at any point beyond a the stress is released, 
it is found that the piece no longer returns to its original dimen- 
sions but has been permanently distorted. 

If at any point on the curve below a the stress be divided by 
the strain a ratio is obtained which is constant for all points 
below a. This ratio is called the modulus or coefficient of 
elasticity. If, therefore, this modulus of elasticity is known for 
a given material, the strain corresponding to any given load may 
be calculated, providing it does not exceed the value correspond- 
ing to the point a. 

The point a is called the elastic limit and is well-defined in 
most materials. Cast iron has, however, no well-defined elastic 




Fig. 6. 



limit and little permanent elongation. Materials of this kind are 
said to be brittle. 

If sufficient tensile stress is applied to a test piece its elonga- 
tion increases until finally it " necks down" at its weakest point 
and rupture occurs. The load per unit area under which a bar 
breaks is called its ultimate strength and the corresponding stress 
or load per unit area is called the ultimate stress. Similar phe- 
nomena are observed when a piece is tested in compression or 
torsion, etc. 

It is evident that the working stress of a machine member 
must be less than the elastic limit if the piece is to retain per- 
manency of form. The stress at which a member is designed to 



STRAINING ACTIONS IN MACHINE ELEMENTS 35 

be operated is called the working stress and the ratio of the 
ultimate stress to the working stress is called the factor of safety. 
It is to be especially noted that not only must the working stress 
in the member be kept below the value where permanent deforma- 
tion takes place, but also so low that the resulting strain, whatever 
it may be, shall be so small as not to destroy the proper alignment 
of the piece, or cause unnecessary friction through distortion. A 
machine member may be amply strong enough to carry the load 
with perfect safety, and yet distort so badly under the load as to 
render it unfit for the service desired. Both strength and stiffness 
should therefore be kept in mind in designing a machine part, as 
sometimes one and sometimes the other will dictate the form and 
dimensions to be used. A short discussion will now be given of 
the relations which exist between load, stress, and strain for the 
cases most often met and of their bearing on the selection of the 
form and size of a machine member. In this discussion it will be 
assumed that the load is a dead load applied without shock, and 
the modifying effect of suddenly applied and repeated loads will 
be considered after the fundamental relations between load and 
stress are established. 

9. Tension. Let p be the stress in the section, P the load, 
and A the area of cross section. The relation which exists 
between them in simple tension is 

p 
P = -J- W 

And if E be the coefficient of elasticity and I the length of the 
member, the total elongation A is given by the equation 

A 

The elongation per unit of length or the strain = y. 

If, then, a tension member is to be designed to join two 
machine parts, the formula for strength dictates a piece of uni- 
form cross section without regard to any particular form. Hence 
the most convenient or cheapest form would be used, avoiding 



36 MACHINE DESIGN 

thin, wide sections where concentrated stress at the edge might 
cause undue strain. 

Suppose it is required to hold the two surfaces within 
certain limits, as is often the case in machine tools where accu- 
racy is desired. If the tension member is long it may yield 
more than is desirable, though the working stress may be well 
below the elastic limit and a greater area may be necessary to 
reduce a to the desired value. 

Example. Let P = 20,000 lbs., let the allowable stress p = 

10,000 lbs., let E — 30,000,000, let / = 4o", and let it be required 

to keep A within .001". If the design is based on allowable stress 

alone, 

P 20,000 

A = — = ■ = 2 square inches. 

p 10,000 

_ PI 20,000 X 40 

But for A = .001, A = — - = — = 26 sq. in. 

A E .001 X 30,000,000 

In general, therefore, where tension members are of any con- 
siderable length and distortion under load is of importance, they 
should be checked as above. 

10. Compression. If the member under consideration be 
subjected to compression, the remarks of the last paragraph 
apply equally well if the member can be considered a short 
column, i.e., one whose length is not greater than six times its 
least diameter. If longer than this it must be considered as a 
long column and the conditions governing its design will be 
more fully treated hereafter. (See Art. 20.) 

11. Shear. If the member is subjected to simple shear the 
expressions for the relations existing between the stress, area, 
and load are similar to those for tension or 

a- 7 ■ (Q 

12. Torsion. If the member is subjected to a torsional stress, 
the following relations exist : 

Let P =load applied in pounds. 
a = arm of load in inches. 



STRAINING ACTIONS IN MACHINE ELEMENTS tf 

Let 7 p = polar moment of inertia of the section in biquadratic 
inches. 

p s = shearing stress in lbs. per unit area at outer fibre. 

e = distance from neutral axis to outer fibre in inches. 

/ = length of member in incheSc 

= angle of deformation in radians, 

T = twisting moment applied to member in inch pounds, 

E s = transverse coefficient of elasticity. 
Then for torsional strength in general, 

P I 
Pa = T== Zi^ (Z)) 



For a circular shaft of solid section, 



(E) 



16 .... 

For a hollow circular section whose outside and inside diameters 
are d t and d 2 respectively, 

T = J6d7~ (F) 

For deformation under stress for a solid circular section, which is 
the most common case, 

= 7eJ* < G) 

and for a hollow circular section, 

32 Tl 

§ r*B.w-v) • {H) 

An inspection of equation (D) shows that the torsional 
resistance for a given stress is proportional to the polar moment 
of inertia divided by the distance from the neutral axis to the 
outer fibre. Examination of equations (E) and (F) shows that 
for circular sections torsional strength is proportional to the 
third power of the outer diameter. Equations (G) and (H) 
show that torsional deformation is inversely proportional to the 
fourth power of the outer diameter, hence torsional stiffness is 
directly proportional to the fourth power of the outer diameter. 

For a given amount of material that section in which this ma- 
terial is distributed farthest from the gravity axis will be strongest 



3» 



MACHINE DESIGN 



and stiffest as long as the walls of the section do not become so 
thin and weak as to yield locally from other causes. The hollow 
circular and hollow rectangular sections, commonly called the 
"box section," Fig. 7, are best adapted, therefore, to resist tor- 





Fig. 7. 



Fig. 8. 



sional strains. The box section is peculiarly useful in machine 
construction, as many machine members must carry a combina- 
tion of stresses. Machine frames may be subjected to tension, 
compression, or shearing, combined with torsion, and the box 
section, while equally good for simple stresses, is, as has been 
noted, vastly superior in torsion. Furthermore, the box section 
is well adapted to resist combined flexure and torsion. The 
flat sides of a box section also afford facilities for attaching 
auxiliary parts and its appearance is one of strength and sta- 
bility. The thickness of the walls being thinner in hollow than 
in solid forms insures a better quality of metal in castings and 
also more skin surface, where the greatest strength of cast iron 
lies. An advantage not to be overlooked in some lines of work 
is the ease with which hollow sections can be strengthened by 
increasing the thickness of the walls by changing the core with- 
out changing the external dimensions. The cost of pattern 
work is about the same, in general, for hollow sections as for I 
or other sections, while the work in the foundry is, in general, a 
little greater. 

Example. A circular cast iron boring bar 60 inches long 
carries a solid circular boring head 60 inches in diameter. The 
bar is subjected to a torsional moment of 60,000 inch pounds 



STRAINING ACTIONS IN MACHINE ELEMENTS 39 

which is applied at one end. It is desired to keep the torsional 
deflection of the tool below -3V when the bar is transmitting 
power through its entire length, in order to prevent chattering 
of the tool. What should be the diameter of the bar if the 
working stress be taken as 3,000 pounds per square inch and E s 
be taken as 6,000,000. 

For torsional strength from formula E, 

, 7Q 60,000 X 16 

d 6 = = 100 

3,000 X- 

.-. d = 4.6". 

1 

For torsional stiffness 0* = — = -giro since e is in radians 

3° 
and the length of an arc = re, where r = radius. 

32 X 60,000 X 60 

. . from G, a* = — = 5,870 

ttX 6,000,000 X 9-fo 

hence d = 8.8". It is evident that the shaft will be amply strong 
if designed for stiffness, therefore the last value would be used. 
If the section is made hollow less metal can be used. In this 
case either the inside or outside diameter or the ratio between 
them can be assumed. Let 

- 2 = i or d = 2_i 
di 4 4 

o, J4 Tie 

whence, d 2 4 = — ^- and d* — d 2 4 = — - d* 
256 256 

Substituting in H, 

d ._^_i7|^ = _3»X 60,000X60 



256 ~ X 6,000,000 X g^-g- 

. *. dx = 8,550 and d x = 9.6", hence d 2 = 7.2". 

The area of the hollow shaft =-31.67 sq. in., while the area of 

* The angular deflection or twist of a shaft in degrees =5 7. 29 6 X (Angular 
deflection in radians). 



40 MACHINE DESIGN 

the solid shaft = 60.84 s q- m -> so that with a small increase in 
diameter one half the metal secures, by using the hollow section, 
the same stiffness. 

13. Compound Stresses. In the cases of simple loading 
just discussed only one form of stress is brought on the member 
and the design of the cross-section can be safely based on this 
stress. When, however, the loads applied induce stresses of 
several kinds, it is no longer possible in general to base the 
design on any one stress, but regard must be had to the combina- 
tion of stresses that may occur. In many cases one or more of 
the stresses are so small, or their action is such, that they may be 
neglected in designing the member, though they should always 
be borne in mind. The stress on which the design of the mem- 
ber is based may be called the predominating or primary stress 
and it may be a simple stress or a combination of simple stresses. 
The latter will be called a Compound Stress. 

14. Flexure. When a beam is subjected to simple bending 
the principal stresses that are induced are (a) a tension on one 
side of the neutral axis, (b) a compression on the other side of 
the neutral axis, and (c) a shearing stress which acts on every 
section of the beam at right angles to the tension and com- 
pression. Generally speaking, the shearing stress is small com- 
pared with the tension or compression and can often be neglected. 
It must never be forgotten, however, and where the beam is 
designed to withstand the bending moment only, care should be 
exercised that the sections which are subjected to a small bend- 
ing moment are not made so small as to yield under shear. The 
predominating stress will be the tension or compression depend- 
ing on the material and the form of section. 

When a beam is subjected to simple flexure, — 
Let M = bending moment at any section in inch pounds. 

I = moment of inertia of section in biquadratic inches. 

e = distance from neutral axis to outermost fibre in inches. 

A = deflection at any point in inches. 

P =load applied in pounds. 

p = maximum stress at outer fibre in lbs. per sq. inch. 

E= coefficient of elasticity. 



STRAINING ACTIONS IN MACHINE ELEMENTS 4 1 

Then for strength, in general, within the elastic limit, 

pi 

M= — * -. . . . ■. . . . U) 
e 

Every beam when loaded deflects somewhat, depending on the 

shape of its cross-section, the material, the way in which it is 

supported, and the load applied. The curve assumed by a beam 

loaded within the elastic limit is called the elastic curve and is of 

course different for different combinations of the above conditions. 

The general equation of the elastic curve, whatever the shape of 

d? y M 
the beam may be, the load, or manner of support is, - — ^ = -=-7. 

(1/ X Lid L 

To find the particular equation for any case, M must be expressed 
in terms of x and the expression integrated twice. The ordinate 
y, which is the deflection, can then be found for any value of x and 
its greatest value is the maximum deflection. This integration 
has been performed for all the cases usually met with in practice, 
and the results are tabulated in Table I. It is to be noted that 
this tabulation is for beams of uniform, section and for stresses 
within the elastic limit. Here, as in other classes of machine 
members, the design of the part may be based on strength or stiff- 
ness, depending on the conditions, and in general both should be 
considered. 

Example. A steel I beam 20 ft. long and supported at the 
ends is used as a track for a crane trolley carrying 4,000 lbs. 
Select a standard rolled / beam that will carry the load with a 
deflection of not more than yV at the centre and a maximum 
stress of not more than 8,000 lbs. 

From Table I, 

3" PI 3 4,000 X 240 3 



A 



16 48 EI 48 X 30,000,000 X / 



4,000 X 240 3 X 16 

whence I = — = 205. 

48 x 30,000,000 x 3 



* The expression — is sometimes called the modulus of the section and is 

generally indicated by the letter Z. It should he noted, however, that this ex- 
pression is applicable only to symmetrical sections as e may have two values for other 

sections. pi. , , . . 

; — is termed the resisting moment. 



42 MACHINE DESIGN 

From handbooks on structural shapes it is found that the 

moment of inertia of a 12" I beam weighing 31.5 lbs. per foot is 

215.8. Let such a beam be chosen. Then from formula J, the 

Me 2,000X10X12X6 „ _, 

stress p = —7- = = 6,700 lbs. nearly. The 

/ 215.8 J 

section therefore is satisfactory. 

15. Beams of Uniform Strength. The values in Table I 
refer to beams of uniform cross-section. In nearly all cases the 
bending moment, which is usually the basis of design, varies and 
if, therefore, the beam is made strong enough at its most strained 
section and uniform in cross-section throughout its length it will 
have an excess of material at every other section.* Sometimes it 
is desirable to have the cross-section uniform, while in other 
cases the metal can be so distributed that every section shall 
have the necessary strength to resist the bending moment and no 
more. In the latter cases the shearing stress must be looked after 
carefully. Table II gives a few of the forms most usually met 
and an example may make their application clear. 

Example. A cantilever of rectangular section 30 inches long 
carries at its outer end a load of 1,000 lbs. It is to have a uni- 
form thickness. What is its vertical outline so as to have uni- 
form strength ? 

Let the thickness = b and the variable height —y. Then the 
moment at any section at a distance x (Fig. I, Table 2) is Px, 
and this must be equal to the resisting moment of the section at 
each point, hence 

pi pbf , 6Px 

Px = L - = Z-f- or f = — — 

e 6 J pb 

which is the equation of a parabola whose vertex is at the outer 
end of the beam. In the problem assumed let b = 1.5 inches and 
let p = 4,000 lbs. Then when # = 30, y — h — $.$". In a similar 
way other points may be found or the curve may be laid out by 
graphical method. The shearing load at any point is P, and 
hence the shearing stress increases as the cross-section of the 

* This of course does not cover the possible case where the effect of shearing 
or other stresses may exceed that due to flexure. 



STRAINING ACTIONS IN MACHINE ELEMENTS 43 

beam decreases. When x = o, v = o, and in general when x 
is small, y is very small; therefore the outer end of the member 
must be modified so as to safely carry the shearing stress. Refer- 
ence will be made to this again under the section dealing with 
machine attachments (see chap. 16). It is to be especially noted 
that these theoretical shapes are based on certain assumptions 
and unless these are observed in the design, the theoretical out- 
lines do not apply. Thus in the cantilever example above, if the 
thickness of the beam is not kept uniform the outline for uniform 
strength is not a parabola. The mistake of using a parabola 
when the thickness is not uniform is often made when I or T 
sections are used instead of uniform thickness or depth. It is 
evident, that, whatever may be the form of section adopted, by 
means of the bending moment and shearing load the correct 
depth of section can be found for a number of points and a 
curve plotted that will answer the requirements of uniform 
strength. 

16. Combined Flexure and Torsion. Let the force P, Fig. 8, 
act upon a rod with an arm a at a distance from the support equal 
to /. Then the stresses induced in the section close to the support 
are 

(a) flexure due to the bending moment PI 

(b) torsion " " " twisting " Pa 

P 

(c) shearing "" " " direct load and equal to — . 

The shearing stress is usually very small compared to that due 
to bending and twisting, and can be neglected; the predominat- 
ing stress therefore is that due to the combined action of the 
bending and twisting moments. 

It can be shown that if a bar or rod is subjected to a longi- 
tudinal tensile or compressive stress and at the same time to a 
shearing stress at right angles to its length, the combination of 
these stresses may produce similar stresses greater than either 
and acting along planes other than those along which the original 
stresses act.* 

* Church's "Mechanics," page 317. 



44 



MACHINE DESIGN 



TABLE I 

BEAMS OF UNIFORM SECTION 



Diagram of Loads, 

Bending Moments 

and Shear 



Greatest 

Bending 

Moment 

M 



Location 

of 

M 



Greatest 
Deflection 



Location 
of 



Maximum 

Shearing 

Force 



Section 

where 

Shear 

is Max. 



B 



^ 




PI 



PI 3 
3 EI 



Any 




Pjli + R,!, 



B 



P1 + P2 



From 
CtoB 



ooooooy 




wl ; 



Wl 3 
8EI 



wl = W 




wr 



+ P1 



ElL3^8j 



'1 + P 



Combination; 
of Curves 
1 and 3 




_ 5 



iz P[ 



+— Pi- 



+ C 
* B 



PI 3 At- 



107 EI 



0.45 1 

from 

A 



iL p 

16 

16 



BtoC 
CtoA 




onofioor ^ 






9w|" 



2 p. 

OS 



128 
wl 



* c 

* B 



wl 3 
185 EI 



3 wl 
8 

5wl 



A 
B 



w=\oa.d Der unit length. W— total distributed load. P= concentrated load. 



STRAINING ACTIONS IN MACHINE ELEMENTS 



45 



TABLE I — Continued 



Section 
where 
She;;r 

i^ Mux. 



Diagram of Loads, 

Bending Moments 

and.Shear 



Greatest 

Bending 

Moment 

M 



Location 

of 

M. 



Greatest 
Deflection 



Location 
of 

A 



Maximum 

Shearing 

Force 



VII 




® s 
£ 8 

« ® 
2 * 

OO 



PI 

2 



AandB 



PI 3 
12 EI 



Equal 
in 
All 



g")Oooooq V 

Ib Ua 



VIII 




® 5 

© 



OO O 



3 

wl 

6 





At B 



At A 

At 

0.4231 

from 

_B_ 



Wl 3 
24 EI 



W 



B 




jfcy 



PI 

4 



pr 



48 EI 



_P 

2 

_P 

2 



BtoC 
ClroA 




o d 



° „- ^ 



Plll 2 
1 



PI 2 1 2 

C *1 *2 



At C 



3 ,» 



31EI 
Not Max. 



BtoC 



CtoA 



05SO 



B C 




CD 

■Sg3 



PA 






PIi 



C to D 



24 EI 

[ 3 1 2 -4 1 2 ] 



Centre 



BtoC 
DtoA 



B ooon c oono A 



XII 




wl' 



Centre 



5 Wl 3 
3b4EI 



Centre 



wl 
2 



AorB 



w=load per unit length. W=-- total distributed load. P= concentrated load. 



46 



MACHINE DESIGN 



TABLE I— Continued 



Diagram of Loads 
Bending Moments 
and Shear 



Greatest 

Bending 

Moment 

M 



Loca- 
tion of 
M 



Greatest 
Deflection 



Loca- 
tion of 



Maximum 
Shearing 

Stress 



Section 
where 
Shear 

is Max. 



B C A 

0000,0000 



XIII 







[P+.^L] J 



(p+fw)x 
I 3 

48 EI 



£+2 

*) ' o' 



Aor B 



P) P 



kaJ 



— F . 



u4 



XIV 1 



c a 



Pa 



DtoC 



BE) E C A 

OOOOOOOOO^ 



9 




CS eg 

3l 






Waf 

2.1 



CandD 



W 



C«ndD 







XVI 



13 

a 

01 

° 3 a 

+? ® o 
a a « 

<D 
M 



PI 



at 
A.B. 
and C. 



pr 



J92EI 



JP 

2 

P. 

2 



BtoC 
CtoA 



B ^ QOQOOOO [| 



XVII 




0) CS 

*3 



wl 
12 



AorB 



wl 4 
384 EI 



2 



AorB 



XVIII 




|>3 



I 2 

I 3 

Pl 2 l 2 f 
I 2 



at B 

atC 

at A 



2 2 

2Pi;i,. 

3EI(l+2lj) 



w=load per unit length. W= total distributed load. P= concentrated load. 



STRAINING ACTIONS IN MACHINE ELEMENTS 



47 



TABLE II 

BEAMS Or UNIFORM STRENGTH 



Outline 

of 

Beam 



Greatest 

Bending 

Moment 

M 



Loca- 
tion of 
M 



Greatest 
Deflection 



Loca- 
tion of 



Maximum 
Shearing 

Stress 



Section 
where 
Shear 

is Max. 




>5 . 

5§ § 

— 1 += .ii 

& o « 

■* i» y 
3 QS © 
O .« 

a * 
C4 



SP1 ; 



PI 



Ebh 



any 



o 

a 




f^Diagram of 
Moments and Shear 
same as I 



© C £ 

30 a? 



C r =f 



oJ a 






Pi 



6Pr 

Ebh a 




Diagrams 
as in I 



« « ^ 
&£ s 



M 
0,1 P. 

II 



any 



|DOOpOOOO( 



Diagrams as 
in No. Ill /Table I 



13 o 

:qp 






wl- 
2 



w 



r— *■-- ^■+ s -*| 

1 Diagrams same 
as No. X Table I 



O °3 

So 
p-g. 

■s o 

S3 & 

2H 



e-o. 1 



Pljlo 



when 1] "=lo 
A = 

PI 3 

2Ebda 



DDDD5SXJXI 




Diagrams same 
as No. XII Table I 



9 £ 

o ;r 

& o 

* -s 

M — 



II 



wl J 

s 



Centre 



VII 




PI 

4 



Centre 



3P1 3 

8Ebd ; 



Diagrams same 
as No. IX Table I 



w=load per unit length. W= total distributed load. P= concentrated load. 



48 MACHINE DESIGN 

If / be the greatest direct tensile or compressive stress and s 
the greatest direct shearing stress applied to the bar, then the 
maximum tensile or compressive stress p due to / and s is given 
by the following equation : 

^ = |[/ + v'F+7?] . . . . (1) 

and the maximum shearing stress p B due to / and s is 

A = I v/ 2 "+7? ( 2 ) 

It is evident that the numerical value of p will always exceed 
that of p a and therefore if the material used has approximately 
the same tensile and shearing strength the design can be safely 
based on (1). But should the allowable shearing strength of the 
material be less than the tensile strength, as is usually the case, 
it may happen that the shearing stress p a as found by (2) would 
dictate a larger section than that required by p as found by (1). 

If the tensile stress is due to a bending moment and the shear- 
ing stress is due to a twisting moment the values of / and s can 
be found from equations J and D respectively and p and p t 
obtained as above in equations (1) and (2) respectively. 

Example. A certain section of a circular cast iron shaft is 
subjected to a bending moment M of 10,000 inch lbs. and a 
twisting moment T of 60,000 inch lbs. The allowable tensile 
stress p, is 2,000 lbs. per square inch and the allowable shearing 
stress p a , is 1,600 lbs. per square inch. It is required to design 
the cross-section of the shaft. 

Me 32 M 32 X 10,000 100,000 

From J, t = — — = „ = -z — = ^ — nearly 

/ t: d 3 - x d 3 d 3 J 



Te 16 T 16 X 60,000 300,000 

77 = Tip = " ~7d 3 



and from D, s = — = -^-^ = --= = — — 3 — nearly 



r , 35°> 000 , • 

hencefrom(i),^, = — —^ — and since p = 2,000 

2,000 
or d = 5.55" 

_ 300,000 . 300,000 
From (2),p = — -v— and since p = 1,600, dr = — ; 



STRAINING ACTIONS IN MACHINE ELEMENTS 49 

.•.^ = 5.8" or %" greater than that given by (1). It is evident 
that the last value should be taken. 

Equations (1) and (2) are general, and applicable to any and all 
sections, but for circular shafts operating under conditions that 
produce both bending and twisting it has been found convenient 
to make use of what may be called an equivalent or ideal bending 
moment which may be derived from equation (1) as follows. 

Let M e = the equivalent bending which will produce the 
maximum direct stress p. 

Let M = the bending moment producing the direct stress /. 
" T = the twisting moment producing the shearing stress s. 
" r = radius of shaft. 

From 7, M = — and M = ■—, 
r r ' 

_ _ s L 2 si 

and from D, T = = 

r r 

(Since J p = 2l for circular or other sections for which the mo- 
ments of inertia about two perpendicular axes are equal.) 

Multiply equation (1) through by — , whence 



Y 2 *~ f \ r 2 r 2 —> 

Pi 

r. — = M t = y 2 M + y 2 v M 2 + T 2 . . . (A) 

In a similar manner an equivalent twisting moment can be 
deduced from (2) thus, 

^y- = T e = VM 2 + f 2 .... (K t ) 

The quantities M and T are usually large and the numerical 
work involved in solving K and K t can be simplified by writing 
M = x T, where x for any particular problem will be a known 
quantity. 

Whence K reduces to, 



^ e = K T [x + V x 2 + 1] . . . . (K 2 ) 
and K x reduces to, 

T. = T V ' * 2 ~+T (iQ 

4 






50 MACHINE DESIGN 

It is to be especially noted that M e and T e are equivalent mo- 
ments in a numerical sense only; that is, if a bending moment 
M and a twisting moment T are applied to a shaft, producing a 
tensile stress t and a shearing stress 5 respectively, then M e is 
a bending moment which will give a stress equal to the maximum 
resultant tensile or compressive stress p, and T e is a twisting 
moment which will give a stress equal to the maximum resultant 
shearing stress p s , reference being made to the same section. 

The application of these equations to the investigation of any 
existing shaft subjected to a bending moment M and a twisting 
moment T is obvious, and it remains to consider their applica- 
tion to the design of new shafts. It has been pointed out that 
the greater numerical value given by equation (i) does not nec- 
essarily indicate that a larger section will result from its adoption 
than would result from the use of equation (2). For the same 
reasons the greater numerical value of M e , obtained from K may 
not give a larger section than would be obtained from T e by ap- 
plying K v It is necessary therefore to determine under what 
conditions each should be used for designing in order that the 
maximum diameter of shaft shall be found in all cases. 

pi pxd 3 



is 



From /, M e = 



3 2 



Whence ^ = ^^X^ ../.'. (3) 



(4) 











71 p 




It 


In 


a 


similar 


way 


from E 

TX 


X 


T, 

A 



Since in any given problem M and T are always known, M m 
and T e can always be found from K and K x (or K 2 and K 3 ) and 
since the allowable values of p and p s can always be assigned, the 
diameter of the shaft d. can always be determined from both 
equations (3) and (4) and the larger value selected as in the problem 
previously solved. It is, however, desirable to know, for any 
set of conditions, whether equation (3) or equation (4) will give the 
greater value of d without the necessity of solving both equa- 
tions. 



STRAINING ACTIONS IN MACHINE ELEMENTS 5 1 

It is evident that in order that equations (3) and (4) may give the 

.. . , f 2 M e T e T e p. 

same diameter of shaft must equal — or — — t — - L - and 

P P. 2l e p 

that for conditions other than these, either equation (3) or equa- 
tion (4) may give the greater diameter. It is therefore neces- 

T p 

sary to investigate the relations existing between — ■— and — 

for three sets of conditions. 

(1) When equations (3) and (4) will give equal values of d. 

(2) equation (3) will give the greatest value of d. 

(1) It has already been shown that equations (3) and (4) will give 

p s T e p s 

equal values of d when — - 8 = — 77- or if — - 8 be called y. then 

p 2M e p " 



T e VM 2 +T> 

y 2 M e M + VM 2 +T* '-•■• (5) 

is the equation of a curve which expresses all the simultaneous 

values of — and -—— for which equations (3) and (4) will give 

equal values of d. The value of either M or T in equation 5 may 
vary from zero to infinity and the most convenient way of plotting 
simultaneous values of M and T is to plot their ratio. If then, 
in equation (5), the relations as given in K 2 and K 3 be substi- 
tuted for those in K and K x the equation becomes 



y = = === (6) 

2 M e x + Vx>+ I 

which is the equation of a curve expressing all the simultaneous 

values of y (or — - S J and x (or — ) for which equations K 2 and 

K 3 will give equal diameters of shaft. 

It is desirable before plotting the curve to examine the limits 
between which x and y may vary. It is clear that for M = o 
x = o, and ior T = o x= 00 , hence the limits of x are o, and 00 . 



5 2 



MACHINE DESIGN 



Using these same limits for M and T in equation (5) it is found 
that 

when M = o, y — 1 and M e = — 

and when T = o, y = % and M e = T e 
That is for all materials where the ratio of allowable shearing 
to tensile stress lies between 1 and ]/ 2 there are always simulta- 
neous values of M and T for which equations (3) and (4) will give 
equal values of d. The curve giving these simultaneous values 
is shown in Fig. 9 and has been plotted from equation (6). 

(2) If for any given value of — within the limits 1 and y 2 a 



Q.JO. 
li 

t».7 



.5 





















Field of K 




































M 


3 = 


























<[a 


+\1 


x. 2 


4- I 


y 


? 


Fi 


eld 


of 


K> 








































i 


I 




































T 


e _r 


rv 


X2 + 



























































.1 .2 .3 .4 .5 .6 .7 .8 .9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.0 2 2.1 2.2 

Fig. 9. 

ratio of — be taken greater than the simultaneous value given 

by the curve (or in other words if the co-ordinates chosen inter- 
sect above the curve) equation (3) will give the largest value of d. 

For the value of — can be increased only by making M greater 

relatively to T and an examination of K and K x shows that in- 
creasing M increases K more rapidly than it does K v Hence 
in such cases K (or K 2 ) applies and equation (3) which is based 
on them will give the largest value. 

Further, for values of — equal to or greater than unity, equa- 

P 
tion (3) will also give the largest value of d. For it has just been 

T 

shown that M can never be less than — e and only equals this 

6 O 



STRAINING ACTIONS IN MACHINE ELEMENTS 53 

when M = o. For all finite values of M> therefore, M t must be 
greater than — ! ; and it is evident from equations (3) and (4) that 

T e 
for values of p a = p and M e > — e - equation (3), which is based on 

K (or K 2 ), will give the greatest value of d. 

(3) In a similar way it can be shown that for all simultaneous 

p s M 
values of — - and — which intersects below the curve and within 
p T 

Ps 

the limits y = 1 and y = /4> or f° r a ll materials where — is less 

than >£, equation (4), which is based on K x (or K 3 ), will give the 
greatest value of d. 

Summary. Equations K 2 and K 3 are the most convenient 
forms of equivalent moments and will be used in this work. It 
is to be particularly noted that they are applicable only to circu- 
lar or other sections where the polar moment of inertia is equal 
to the sum of the rectangular moments of inertia around perpen- 
dicular axes (see page 50). Where the section to be considered 
is more complex the solution must be based on the original equa- 
tions (1) and (2) in a similar manner to that employed in the ex- 
ample on page 49. Equation K 2 should be used where the 

simultaneous values of * and f mterS ect above the curve which 

p T 

Ps 

is always the case whenever — -> 1. Equation K 3 should be 

used where the simultaneous values of — ! and — intersect below 

p T 

Ps 

the curve, which is always the case whenever — 1 <J. 

Example 1. An engine cylinder is i6 // X24 ,/ (piston 16" in 
diameter and stroke of 24"), steam pressure = 100 lbs. per square 
inch. The centre of the crank pin overhangs the centre of the 
main journal by 15" (measured parallel to the axis of shaft). 
Assume that the pressure on the crank pin may be equal to 100 
lbs. unbalanced pressure per square inch of the piston when 
the connecting-rod is perpendicular to the crank radius. Allow- 



I 



54 MACHINE DESIGN 

ing 8,000 pounds as the maximum allowable direct stress and 
6,400 as the maximum allowable shearing stress, compute the 
diameter of the shaft. 

Area of piston = 200 sq. inches; radius of crank (arm of 
maximum twisting moment) r = 12"; arm of bending moment 

= 15" 

T = 200 X 100 X 12 = 240,000 inch lbs. Also 
•M = 200 X 100 X 15 = 300,000 

M p a 6,400 

x = — = it; — 12 = 1.25; — = r = .8 = 7. 

T D P 8,000 7 

p a 
By referring to Fig. 9 it is seen that for y = — - = .8 and x = 

1.25 the ordinates intersect above the curve, hence K 2 should be 
used. 



From K 2 , Af e = J [1.25 + \/ 1.25 2 + 1] 240,000 

= 342,000 inch lbs. 

^ , s ,0 ^42,000 X 12 

From ( 3 ), * = 6 \ ' j = 435 

5,000 X n 

.-. rf = 7-58" 

Example 2. — A circular cast iron shaft is subjected to a twist- 
ing moment of 250,000 inch lbs. and a bending moment of 62,500 
inch lbs. The allowable tensile stress is 2,000 lbs. per sq. inch 
and the allowable shearing stress 1,400 lbs. Determine the 
diameter of the shaft. 

Here y = — - = — = .7 and x = — — — = .2<. 

p 2,000 250,000 

. p a ■ 

From the curve, Fig. 9, it is seen that for y = — = .7 and 

x == .25 the intersection of the ordinates falls below the curve, 
hence K s should be used. 



Then T e = [VO5) 2 + 1] 250,000 = 257,500 
16X257,500 

re X 1,400 " 935 

d = 9.75 inches. 
Suppose, however, that K 2 should be used. 



STRAINING ACTIONS IN MACHINE ELEMENTS 



55 



Then, M e = \ [.25 + \/(.25) 2 + 1] 250,000 = 160,000 

# = 32 x l6o ' 00 ° - 830 

- X 2,000 
.*. d = 9.4 inches or .35" less than the value given by K 2 . 
A convenient graphical solution of K 2 and K 3 is shown in Fig. 
9 (a) which may be used as follows: 

For K 3 make Oa = unity; lay off Ob = x to the same scale on 
the vertical axis. Draw ab extending it beyond b for a 
length somewhat greater than x; 

then ab = V Ob 2 + Oa 2 = Vx 2 + 1 ; 
hence T e = ab X T. 
For equation K 2) lay off be = Ob = x along the extension of 
ab. Then be + a b = ac = x + V^ 2 + 1 ; 
Hence M e = ]/ 2 [ac X T]. 




Fig. 9 (a). 

17. Other Formulas. — Equation K is sometimes trans- 
formed into an equivalent twisting moment. Since in general 

pi 2 p s I 

M = — and T — — — , for an equal intensity of stress (that 

i s j p B = p) T = 2M for the same section. If therefore it is con- 
sidered more convenient to use an equivalent twisting moment 
instead of an equivalent bending moment it is allowable to sub- 
stitute for M e (the bending moment, equivalent to the combined 
bending and twisting moment), y£ T e (a twisting moment equiva- 
lent to the combined bending and twisting moments) provided 
the same allowable direct stress is used with T e in solving for the 
diameter of shaft. 

.\ T e = 2 U = M + sjWTT <.'..-. (K 4 ) 



56 MACHINE DESIGN 

Equations K 2 , K 3 , and K 4 are all different forms of Rankine's 
formula for combined bending and twisting. Other authorities 
give slightly different coefficients. Thus Grashof gives 

M e = | M + f VM 2 + V .... (7) 
While others give 

M e = 0.35 M + o .65 VM 2 + T 2 . . (8) 
The diameter of shaft given by equations (7) and (8) will 
not differ much from that given by K 2 , for any set of conditions, 
except where the bending moment is very small. At the limit 
where the bending moment M is equal to zero, Grashof's formula 
gives a value of M e , 25 per cent greater than that given by K 2 . But 
it may be noted that in general for all materials whose shearing 
strength is less than their tensile strength (and this is the case 
for most materials used in engineering) that when M is small or, 
in other words, when the shearing stress predominates, it is 
safer to use K 3 in preference to K 2 . It will be found that for 
the range where equations (7) and (8) give values greater than 
K 2 , that these values will still be less than those obtained from 
K 3 or at least not enough greater to warrant the use of a different 

Pu 

formula in place of K 3 . Take for example steel where — = .8 

P 
and x = .1, which is down close to the limit where Grashof's 

formula gives the greatest value compared to K 2 . Expressing d 3 

in terms of T as in equations (3) and (4) , 



d = 1.77 ^1 
from K 3 d = 1 . 84 ■%] — 



from K % 

*' P 



P 

from Grashof's formula d = 1.88 \ — . 

\ p 

from which it is seen that the difference between d as determined 
by K 3 and Grashof's formula is negligible. The same evidently 
applies to equation (8) which differs but little from Grashof's. 
As the value of x increases, the difference between these equiva- 
lent bending moments decreases, and any variation is more than 
covered by the factor of safety which must be used. 



STRAINING ACTIONS IN MACHINE ELEMENTS 57 

18. Combined Torsion and Compression. Propeller shafts 
of steamers and vertical shafts carrying considerable weight are 
subjected to combined twist and thrust. The span, or distance 
between bearings, is frequently so small that the shaft may be 
considered as subjected to simple compression, so far as the 
action of the thrust is concerned. 

The intensity of this compressive stress in such cases is 

C " vd 2 ' 

in which P = the thrust, and d = the diameter of the (solid cir- 
cular) shaft. 

If T = the twisting moment on the shaft, r = % d = the 
radius of the shaft, / = the polar moment of inertia = 2 I 
(= 2 times the rectangular moment of inertia), and 5 = the 
intensity of shearing stress due to T, then 

r d " ; 4 / ~ w d a 

for solid circular shafts. 

The resultant maximum stresses are those due to the com- 
bined actions of a normal stress (compression) and a tangential 
stress (shear) as in the case of combined bending and twisting 
(Art. 17); hence equations (1) and (2) of the preceding article 
apply and may be used to find the maximum compressive or max- 
imum shearing stress; or if c be the compressive stress due to P, s 
be the shearing stress due to T, p c the maximum resultant com- 
pressive stress, and p the maximum resultant shearing stress, then 



p Q = -c + ~vV -f 4 s 2 and p t = - Vr + 4 s 2 
2P 1 II6P 2 4 (i6) 2 r 2 1 I16P* 4Ti6y 2 ~y 



or^^- 2 [p + ^ + ^] W 



58 MACHINE DESIGN 

It is difficult to find the value of d for a given value of p c or p 
from the above equations, and it is much more convenient to 
assume a trial diameter d and then check for the values of p c 
and p & to see that they do not exceed the allowable compressive 
and shearing stresses of the material under consideration. 

If the span of the shaft between bearings is so great that the 
shaft must be considered as a column likely to buckle, the trial 
diameter of the shaft may be taken so as to bring the mean 
compressive stress c well below the allowable value, and after 
solving for p c and p % the shaft may also be checked as a long 
column (Art. 20). In steel shafting it is necessary usually to 
apply equation (L) only, but it is well to check the shearing 
stress p a against the allowable stress by applying (Z,,). 

19. Flexure Combined with Direct Stress. If the section 
XY, Fig. 10, be acted on by a force P at a distance from its 
gravity axis O equal to a, the stresses induced in the section will 
be:— 

(a) A uniformly distributed stress due to the load P and 

2 

equal to — per unit area. This will be tensile or compressive, 

depending on the direction of P. 

(b) A flexural stress due to the bending moment P a. This 
flexural stress will be a tensile stress on one side of the gravity 
axis which is at right angles to a, and compressive on the other. 

If the direct stress induced in the section by the load P is 
tensile, then the flexural stress on the side toward the load is 
tensile. If the direct stress induced is compressive, the flexural 
stress on the side toward the load is compressive. The maxi- 
mum stress will be the greatest algebraic sum of these combined 
stresses at the outer fibres at X or Y. The distribution of these 
stresses for both cases is shown graphically in Fig. 10, where 
tensile stresses are plotted above the line U V and the compress- 
ive stress below; the ordinates under r s representing the flexural 
stresses, and those under m n the direct stresses. An inspection 
will show where the algebraic sum is greatest. In the case shown 
the combined compressive or combined tensile stresses at X are 
the greatest which may come on the section, depending on the 



STRAINING ACTIONS IN MACHINE ELEMENTS 



59 



direction of P. This is not necessarily so, as a brief reflection 
will show that if O be located near enough to X the reverse of 
the above conditions may exist. The form of section and location 
of the gravity axis should be fixed with reference to the relative 
tensile and compressive strength of the material used. 

Let p r = the direct stress due to P, 

Let p" = the tensile or compressive stress due to P a. 

Let p = the maximum stress in the section at X or Y, 

P P ae 

Then from formula A, p r = — , and from formula/, ft* — — — 

■A. 1 



P P ae* 
Therefore, p = p' + f = — + — y— 



(Af) 




Fig. io (a). 



Fig. io (b). 



where e is the distance from O to the outer fibre at either X or 
F, depending on which is under consideration. If the material 
used is equally strong in tension and compression the gravity 
axis should not be far from central, but where cast iron is used 
it is advantageous to distribute the metal more toward the 
tension side, thus drawing the gravity axis toward that side. 
This increases e on the compression side, and hence increases the 



*See Church's "Mechanics," p. 362. 



6o MACHINE DESIGN 

compressive stress. It decreases e on the tension side, and hence 
decreases the tensile stress. Cast iron is much stronger in 
compression than in tension, and therefore a greater moment can 
be withstood by a given cross-sectional area when distributed in 
this manner. 

It is not practicable, in general, to solve equation (M) for the 
direct determination of the dimensions of a cross section to sus- 
tain a given eccentric load P with an assigned intensity of stress 
p, because both A, I, and e are functions of the required dimen- 
sions; and with any but the simplest sections complicated func- 
tions result. With solid, square, or circular sections, or in 
general where only one dimension is unknown, it is possible to 
reduce M to a form which can be solved; but the algebraic ex- 
pression is a troublesome cubic equation. The practical way is 
to assume a trial section and check this for P or p. 

Example i. A small crane (Fige n) has a clear swing of 28 
inches. The section at m n is shown by Fig. 11 (b). Find the 
load corresponding to a maximum fibre stress (compression) of 
9,000 lbs. per square inch at n. 

P Pae n pAI 

+ —r~ - c . P = 



r A ' / I + Aae 

a = 28 + 2 = 30 A = 2 X 4 — 1-5 X 3. = 3.5 
/ = _i_(2 X 64 - 1.5 + 27) = 7*3 

a p== 9,000X3*5X7 -3 = I?o6olbs . 
7<3 + 3-5X3°X2 

Example 2. A punching machine (Fig. 12) has a reach of 
22 inches. Maximum force P acting at the punch is taken at 
70,000 lbs. Design the section m n so that the maximum fibre 
stress at n (tension) shall be about 2,400 pounds per square inch, 
and check the compressive stress at m. . 

The general form of section best adapted to this case is that 
shown in Fig. 12 (b). Taking the trial dimensions as in Fig. 12 (b), 
the neutral axis is found to be 8" from n. 

.°. a — 22 + 8 = 300 It is also found that A = 216 and 
I = 7.680. 



STRAINING ACTIONS IN MACHINE ELEMENTS 



61 



. • . at n, 

70,000 70,000 X 30 X 8 
P = —Z7Z- + Z7^. = 325 + 2,200 = 2,525 lbs. 



216 
and at m, 

A = 7— 



7,680 
70,000 x 30 X 11 



«6 ' 7,680 =-325 + 3,000 = 2,675 lbs. 

The tensile stress is slightly greater than the limit assigned. 
If this excess is not considered permissible a stronger section 
must be taken. It is evident that this can be accomplished by 
massing the metal still further toward the tension side as the 
compressive stress is very low. 




Fig. 11. 



Fig. 12. 



The theory regarding the position of the neutral axis given 
above is that in general use for such cases. Recent investiga- 
tions have pointed out the fact that this theory is not absolutely 
rigid for curved beams. For the usual case of design it is be- 
lieved that the above is sufficiently accurate. 

20. Stresses in Columns or Long Struts. When a short 
bar is subjected to an axial compressive load the stress induced 
in each section is simple compression (see Art. 20), and the value 
of the stress p is given by formula (^4) or 

W 

If, however, the bar is more than 4 to 6 times as long as its least 
diameter, the above equation does not apply, as the bar will, if 



62 MACHINE DESIGN 

proportioned as above, deflect laterally under the load and will 
ultimately break under a compound stress due to compression 
and lateral bending. Such a member is called a column. 

Theoretical equations for the design of columns were first 
developed by Euler. Other formulae were later developed ex- 
perimentally by Hodgkinson and Tredgold. Gordon and Rankine 
have also proposed equations for the design of this class of mem- 
bers. The student is referred to any good treatise on the Me- 
chanics of Materials for a fuller discussion of these expressions 
than can be given in this work. 

Let I = the length of the column in inches, , — 

p = the least radius of gyration of cross-section = -vl — > 
/ = the least moment of inertia of cross-section, 
A = the area of the cross-section in square inches, 
P c = the breaking load on the column in pounds, 
p' = the mean intensity of stress under the breaking load, 

or the unit breaking load, = P c -f- A. 
p a = the crushing strength of the material, or unit stress 

at the yield point. This is the maximum intensity 

of stress in the column when the mean intensity of 

stress is p e ' y 
n = the factor of safety, 

P = the working load on the column in pounds, P c -*■ n, 
P' = the mean intensity of working stress, or unit working 

load, = pi -f- n = P -r- A, 
p = the intensity of working stress in the column 

( = p c -5- n). This is the maximum intensity of 

stress in the column when the mean intensity of 

stress is p '. 
m — a coefficient for the end conditions as shown in 

Table 3. 
Then Euler's formula for long columns is 

n 2 EI 

It is to be especially noted that Euler's equation is rational and 
ieduced from the theory of elasticity. The coefficient m is also 



STRAINING ACTIONS IN MACHINE ELEMENTS 



63 



rational and applicable to other forms of column formulae. As 
will be shown later, the equation is strictly applicable only to 
very long columns. 

Very short compression members, of ductile material, fail 
under stresses corresponding to, or only slightly in excess of, the 



TABLE III 



'VALUES OF m FOK DIFFERENT EIND CONDITIONS 



CASE I 

Fixed at one 

end, free at 

other. 



m = \ 



CASE II 

"Pin Ended" 

Both ends free, 

but guided. 



CASE III 

"Pin & Square" 

One end nixed, 

the other 

guided. 

m = »/« 



CASE.IV 

"Square 

Ended" 

Both ends 

fixed. 







WW 



"WW/// 



apparent elastic limit, or yield point; for when this stress is 
reached the metal flows, although it does not actually break. 
Very long columns may approximate the resistance as given by 
Euler's formula. Columns of lengths intermediate between 
compression members which yield by simple crushing and those 
which fail by pure flexure are weaker than the former and 
stronger than the latter. If a column is initially exactly straight, 
perfectly homogeneous, and subjected to an absolutely concentric 
load (that is, if it is an ideal column) there seems to be no reason 
why its strength should diminish rapidly with an increase of 
length, other conditions remaining the same. 

However, even an ideal very long column would reach the con- 
dition of unstable equilibrium when subjected to a certain critical 
load (the greatest load consistent with stability). If the load is 
increased beyond this limit and a deflection is caused in any way, 
the deflection will increase until the stress due to flexure pro- 
duces failure of the column. If a deflection is caused while the 
column is under a load less than this greatest load consistent 



64 MACHINE DESIGN 

with stability, the elasticity of the material tends to make the 
column regain its normal form. Initial defects in the form or 
structure of a column or eccentric application of load tend to pro- 
duce such a deflection; hence long struts fail under smaller loads 
than short struts of similar material and cross-section, for the 
ideal conditions are not realized in practice. Or, in other words, 
for equal safety under a given load long columns must have a 
greater cross-section, and lower mean, or nominal, working 
stress.* Even in columns of moderate length, if of ductile ma- 
terial, the flow at the yield point causes buckling. 

Merriman says that if the length of a compression member be 
only from four to six times its least " diameter," it may be treated 
as one which will yield by simple compression. Johnson gives 
limits within which the Euler formula should not be applied as 
I -f- p = 150 for pin-ended, and = 200 for square-ended columns. 
Other authorities give somewhat different limits; but nearly all 
agree that most of the columns in ordinary structures and ma- 
chines are intermediate between simple compression members 
and those to which Euler' s formulae apply. There have been a 
great many column formulas proposed. A graphical represen- 
tation of several of these formulae is shown in Fig. 13. In this 
diagram, abscissas represent ratios of the length of column to 
the least radius of gyration of the cross-section, and the ordinates 
represent the nominal (mean) intensity of compressive stress. Or, 



x = I + P = I + V I +.A, and y = p' e = P c -f- A. 

The diagram is drawn for the ultimate resistance of pin-ended 
columns with a material having a crushing resistance, p c (yield 
point) of 36,000 pounds per square inch, and a modulus of elas- 
ticity, E, of 29,400,000. The value of p\ is 36,000 for a very 
short compression member, and it is evident that a long column 
could not be expected to have a greater strength; hence no for- 
mula should be used which would give a value of p c ' in excess of 

* Owing to the flexure of the long column, the stress is not uniform across the 
section. The maximum intensity of stress must be kept within the compressive 
strength of the material; hence the mean stress is less than for shorter compression 
members, in which the mean stress is more nearly equal to the maximum. 



STRAINING ACTIONS IN MACHINE ELEMENTS 



65 



the crushing resistance p c . Referring to the diagram, it will ap- 
pear that the Euler formula (represented by the curve E E t E 2 ) 
cannot apply to pin-ended columns (of this particular material) 
in which I -i- r < go. If columns with a ratio of / to /> less than 
this limit yielded by simple crushing, and those with a greater 
ratio oil top followed Euler's formula, the straight line F F x and 
the curve F x E t E 2 would give the laws for all lengths of columns. 
It is not reasonable to expect such an abrupt change of law in 
passing this limit (/ -j- p = 90) ; and, as already stated, columns 
of moderate length fail under a mean stress considerably less than 
the simple crushing resistance of the material; or the strength 



F 



30 



20 



® 
£ 10 

CO 

8. 











\ 


































A 




























T^ 






\ 


































s > 


L 
































NjS 




































s V 
































\ H 
































R.N 


NR^ 






































V 








































































































R\ 


































V^i.2 


































T^ 






















































































yN 












































x = 


1-J-p 




















R 2 



100 



200 



300 



Fig. 13. 



of columns is inversely as some function of the length divided 
by the "least diameter." 

Mr. Thomas H. Johnson has developed a formula which is 
based on the assumption that the strength of the column may be 
taken inversely as 7-4- p. This expression is 



in which the coefficient k has the value. 



(1) 



k = 



3 ^ 3 m if 



E 



66 MACHINE DESIGN 

This formula is represented by the straight line T H J 2 in Fig. 
13. It will be noted that this line is tangent to the Euler curve at 
J 2 , and the equation of the latter is to be used, should the columns 
exceed the length corresponding to this point of tangency 
(I ~p > 150). This expression is very simple, after k has been 
determined. It is very convenient in making a large number of 
computations for columns of any one material, and it is employed 
in structural work to a considerable extent. It does not appear 
to have any advantage, on the ground of simplicity, when some 
particular value of k does not apply to several computations. 

For determination of nominal working stress, p' (as computed 
above) may be divided by a suitable factor of safety, n. Or if 
p' -r- n — p f > the expression may be put in the following form for 
direct computation of mean working stress. 



f-±-2-,->-J^* t l - ..•/(■> 



n n ft 3 > 3W- E i) 

Professor J. B. Johnson has derived a formula from the results 
of the very careful experiments of Considere and Tetmajer. 
His formula is: 

>■'-'• ^(7)' (3) 

for pin-ended columns. The curve FB J x (Fig. 13) represents 
this expression. This curve is a parabola tangent to the Euler 
curve, and with its vertex in the axis of ordinates at F, the direct 
crushing stress of the material. For columns having I -r- p 
greater than the value corresponding to the point of tangency J t 
(should such be used), the Euler formula is to be employed. 
The formula of Professor Johnson's is empirical, but it agrees 
remarkably well with very refined experiments on breaking loads. 
It gives considerably higher values for allowable stress' than other 
generally accepted formulas, probably because it is based upon 
more refined tests, or upon conditions further removed from 
those in practice. 

Professor Johnson says ("Materials of Construction," pages 
301-302) that both Bauschinger and Tetmajer "mounted their 



STRAINING ACTIONS IN MACHINE ELEMENTS 67 

columns with cone or knife-edge bearings at the computed gravity 
axis, while M. Considere mounted his with lateral screw adjust- 
ments, and arranged a very delicate electric contact at the side 
so as to indicate a lateral deflection as small as 0.001 mm. He 
then applied moderate loads to the columns and adjusted the end 
bearings until they stood under such loads rigidly vertical, with 
no lateral movement whatever."* 

It would appear that this precaution tends to make the test one 
of the material and not of a long strut; for the eccentricity of 
the load (relative to the nominal geometric axis) compensates, in a 
measure, for the lack of homogeneity of the material. Had the 
correction been made under greater load, the results of the tests, 
if plotted in Fig. 13, would probably be still nearer the line F F u 
and the difference between these test columns and columns as 
used in practice would be greater, requiring a higher contingency 
factor in the latter for safety. 

For determining the working stress, the value of p e ' (as com- 
puted -from the above form of Johnson's expression) should be 
divided by a suitable factor of safety n. Or, the formula may be 
put in the following form for computing nominal working stress : 

The Rankine or Gordon formula (see Church's " Mechanics," 
pages 372-376) has been extensively used for columns. It may 
be expressed as follows : 

P p. 

p: = - a = — f-TTT-, ... (5) 

A I + — 

m 



>Q 



The above formula is based upon experiments on the breaking 
strength of columns. The coefficient /? is purely empirical, and 
this fact limits its usefulness, for it leaves much uncertainty as 
to how this coefficient should be modified for materials different 
from those which have been actually tested as columns. The 

* " This precaution is essential to a perfect test of the material. . . . Only in 
this way can other sources of weakness be eliminated." — [J. B. J.] 



68 MACHINE DESIGN 

mean intensity of working stress, p', might be inferred by divid- 
ing p c f by n, or the expression can be written : 






P' = x //Ti .... (6) 

— /? (- 



but it is not entirely satisfactory to assume the action for stresses 
within the elastic limit, from the results of tests for breaking 
strength. The form of the Rankine expression is rational, but 
the coefficient P is not. 

Professor Merriman says, in his " Mechanics of Materials," 
page (129) : "Several attempts have been made to establish a for- 
mula for columns which shall be theoretically correct. . . . The 
most successful attempt is that of Ritter, who, in 1873, proposed 
the formula 



p p_ 

— 



m r? E V p 



"The form of this formula is the same as that of Rankine's 
formula, . . . but it deserves a special name because it com- 
pletes the deduction of the latter formula by finding for /? a value 
which is closely correct when the stress p does not exceed the 
elastic limit p e ." The above notation is changed to agree with 
that previously used in this article. The ratio p c -5- p is the factor 
of safety. For ultimate strength, this formula might be written: 



P c p c _ 



mr?E \p 



but the first form (eq. N) is the more important. The curve 
R t T R 2 (Fig. 13) is the graphical representation of the last 
expression, eq. AT 1 .* 

Merriman gives the Euler formula for a factor of safety of n = 
P c "*" Pi which is 

*"h-j~* E ($ • • • • (9) 

* Professor Merriman developed equation N\ independently, but later than 
Ritter. He gives Ritter sole credit for the formula in the 1897 edition of his 
"Mechanics of Materials." 



STRAINING ACTIONS IN MACHINE ELEMENTS 69 

Failure occurs if p > £.. The Ritter formula (eq. N) reduces 

to this last expression for columns so long that the term unity 
in the denominator is negligible; strictly speaking, this is only 
the case when I -5- p = infinity. Professor Merriman also shows, 
mathematically, that the two curves, E E x E 2 and R X T R 2 , are 
tangent to each other when I -r- p = infinity. 

If l + p = o, the Ritter formula reduces to p'=P+A, which 
is the ordinary formula for short compression members. 

The facts that this formula is rational in form, that it gives the 
correct values at the limits / -- p = 00 and l + p = o, and that it 
lies wholly within the boundary F F x E x E 2 (Fig. 13), all justify its 
use, and it will be adopted in this work. It will be noted from 
Fig. 13 that the Ritter and Rankine formulas agree very closely 
for the material taken for illustration; but the fact that the 
curve of the latter crosses the Euler curve near the right-hand 
limit of the diagram indicates that its constant /3 is not theo- 
retically correct. 

Exception may be taken to the use of the Ritter formula for 
cast iron, since it involves the use of the stress at the elastic 
limit, and the coefficient of elasticity, both of which have no 
definite fixed values for cast iron. But the same criticism ap- 
plies to the use of any rational formula founded on the elastic 
theory, as far as cast iron is concerned. Thus the expres- 
sions for deflection in simple beams contain E which, for cast 
iron, may vary from 15,000,000 to 20,000,000. Since cast-iron 
columns designed simply for strength are very rare in machine 
design it therefore seems best to use the formula since otherwise 
it fulfils all needs better than any other. 

If it is desired to design a cast-iron column with great accuracy 

values of r-= may be taken which will give results in ac- 

cordance with experiment and which practically transforms the 

P 
equation into Rankine' s formula. If =-= = q, then for cast- 

iron columns with fixed ends q = , for one end fixed and the 

5,000 



7° 



MACHINE DESIGN 



other free but guided q = 
4 



1.78 



, and for both ends free but guided 



= 



5,000 



5,000 
In addition the student should consult treatises on 



the strength of materials treating fully of this subject. 

All of the above formulas give the value of the mean ultimate 
stress (pi = P c -v- A), or the mean working stress (p' = P + A), 
corresponding to a maximum ultimate stress p c or a maximum 
working stress p, respectively. However, the ordinary problem 



11 







2 


4 




Stress in 1000 Lbs. = 
6 8 


10 


12 


14 


9 
8 
7 
6 
5 
4 
3 
2 
1 














/ 






7 


A 




























4 


























N/ 






// 
























/ $ 












IV. M= 4 






// 






// 


/^ 














III. 


M*% 






// 






z> 




s^s 


» 














/ 1 


/// 






/A 


















j.] 


1-1 


i 


/vy 


/v% 


f 
























1 




^5^* 












, 










I.H 


= X 


m 












x = 


+P 

















10 20 



40 



60 80 
Fig. 13 (a). 



100 



120 



140 



of design is to assign proper dimensions for the member under 
the given load. It is not practicable to solve directly, for the 
area in such expressions as those given in this article as p' (or p) 
and p are both functions of the area of the cross-section. It is 
usual to assume a section somewhat larger than that demanded 
for simple crushing, and then to check for the ultimate load P, 
or the working load P' . Professor W. N. Barnard has devised 
a diagram which is very convenient for these computations for 
steel or wrought-iron columns. It is shown, to a reduced scale, 
in Fig. 13 (a). The four curves are for the four end conditions 



STRAINING ACTIONS IN MACHINE ELEMENTS 7 1 

given in Table III, page 63. They are plotted for a maximum 
working stress of 10,000 pounds per square inch, and a value of 

Ks __ > which is an average value for steel. The curves 

E 29,500,000 

should not, however, be used for cast iron, wood, or other mate- 

rials where the ratio ^ will give values far different from the 

above, but such cases may be solved directly by equation N. 
They may be used for any other stress by proceeding as follows: 
Assume a trial cross-section, which fixes p. Divide / by this 
value of p; take this quotient on the lower scale and pass directly 
upward to the proper curve for the given end conditions; then 
pass horizontally to that one of the radiating diagonals which is 
numbered to correspond with the selected stress; from this last 
point pass upward to the horizontal scale at the top of the diagram, 
where the value of the unit load or mean working stress (p f ) is 
read off.* If this value of p' agrees sufficiently well with the 
quotient of the load divided by the trial area, the section may be 
considered as satisfactory. 

In the case of a square-ended column, or when the supporting 
action of the ends is equal in all possible planes of flexure, it is 
sufficient to take the least radius of gyration of the section; or 
to take p for the axis about which the section is weakest. In 
case of a pin-ended column, as a connecting-rod, the cylindrical 
supporting pins make it equivalent to a square-ended column 
against flexure in the plane of the axes of the pins, provided these 
bear symmetrically with reference to the axis of the column ; while 
the column is pin-ended with reference to a plane perpendicular 
to the axes of the pins. If the cross-section of such a column 
has equal dimensions in these two planes (circular, square sec- 
tions, etc.), the column need only be computed for the latter 
plane. If the pin-ended column has an oblong section (elliptical, 
rectangular but not square, I section, etc.), it may be weaker in 



* The method of using the diagram is indicated by the arrows, for an example 
in which l-Z-p=8o and the maximum working stress= 14,000 (pin-ended). In this 
case, p' is found to be about 7,900. 



72 MACHINE DESIGN 

either of these two planes, notwithstanding the difference 
in end conditions relative to them; and it may be neces- 
sary to compute for both planes, unless the section is ob- 
viously stronger in one of them. If a rectangular, or elliptical, 
column has a section in which the dimension in the plane 
of the pins is more than one-half the dimension in the 
plane perpendicular to the pins, it will suffice to compute as 
a pin-ended column against flexure in the latter plane, and 
vice versa. 

In the preceding discussion, the various formulae have been 
given both for breaking and for working loads. The Euler and 
Ritter formulas are derived from the theory of elasticity; hence 
these are proper for computations pertaining to working loads, in 
which the stress should never exceed the elastic limit.* It 
does not follow that these two rational formulas will agree with 
experiments on the ultimate resistance of columns or for materials 
which do not follow Hooke's law of proportionality of stress to 
strain. These expressions are, in this respect, like the common 
beam formulae. Such formulae as Rankine's and J. B. Johnson's, 
derived from tests of ultimate resistance of columns, are, for 
similar reasons, less rigidly applicable to working loads and 
stresses. 

Example. The connecting-rod of a steam engine is 5 
feet long and is subjected to a load of 20,000 lbs. If the 
maximum allowable stress is 9,000 lbs. per sq. in., deter- 
mine the diameter of a circular section at the centre of the 
rod. Take £ = 30,000,000, and the elastic limit £. = 36,000 
lbs. per sq. in. The rod *may be considered a pin-ended 
column. Hence w=i. 

If the rod were designed as a short column, the required area 

20,000 .. r 1 1 

would be A = = 2.2 sq. ins. or a diameter of i-f+ 

9,000 

inches; and it is evident that for a long column the diameter 
must be greater than this. Assume 2K inches as a trial diameter. 

* The Euler formula is not applicable for practical applications, except for 
quite long columns. 



STRAINING ACTIONS IN MACHINE ELEMENTS 73 

d 5 
Then A =4.9, p = - = - / = 60", whence in iV 

4 o 



/ = 



9,000 



1 + 



36,000 



I X 7T 2 X 30,000,000 5 



top = 4,3 °° 



8 J 



.•^ = 4,300X4.9 = 21,070 lbs. which is a little more than the 
required load and the section will fulfil the requirements. The 
student should also follow the solution through on the diagram. 
21. Eccentric Loading of Long Columns. In the preced- 
ing discussion of columns it has been assumed that the load has 
been applied axially. This is obviously the best way of applying 
the load, but cases often occur where it must be applied at a dis- 
tance a from the axis of the column. In such a case the column 
is said to carry an eccentric load, and the arm a is called the 
eccentricity. If the length of the column be less than 4 or 6 

I 

times its least diameter, that is, if the ratio — be less than about 

P 

25, the member may be treated by the method outlined in para- 

P P ae 

graph (10) and formula M will apply or p =— + — - — . 

Jx 1 

If, however, the column be longer than 4 to 6 times its least 

P 

diameter, it can no longer be assumed that the direct stress — - 

A 

due to the load is uniformly distributed over the section, as it 

has been shown by the discussion on long columns that such is 

not the case. 

In addition, if the load is applied eccentrically, it is obvious 
that the column will deflect somewhat more than it would if the 
load were applied axially. This will have the effect of adding to 
the original lever arm a an additional amount a, due to this de- 
flection. 

The stresses therefore acting on an eccentrically loaded 
column are — 

(a) A compressive stress p v such as would be induced if the 
load were axial. 



74 MACHINE DESIGN 

(b) A flexural stress p 2 , due to the eccentricity and propor- 
tional to the bending moment P (a + «). 
For the first from Ritter's formula (N) 

* = 'ft 1 + AeOI 

and for the second from (/) 

P (a + ") e P (a + «) e 

p2 = 7 = a 2 

I A p 2 

Therefore the maximum compressive stress* in the section is 

For columns whose ratio of — is less than ioo, and working 

stresses such as must be used in machine design, the deflection 
a may be neglected. For columns longer than this, or where the 
stress is necessarily high, a can be determined by the theory of 
elasticity. For a full discussion of the manner of computation 
see Merriman's " Mechanics of Materials," 1905 edition, page 
217. For the ordinary cases of machine design this refinement 
may be omitted. 

Example. A circular wooden pole 30 feet high is required 
to carry a transformer weighing 800 pounds, with an eccentricity 
of 10 inches. What must be the diameter at the middle in order 
that the stress due to this load shall not exceed 500 pounds per 
square inch? Let £. = 3,000 pounds per square inch and £ = 

1,500,000. Also m = — . (See Table III.) 

4 

* The stress induced on the convex side of an eccentrically loaded column may 
be either tensile or compressive, but will always be less than the stress on the con- 
cave side. For materials whose elastic strength is about the same in either tension 
or compression the stress on the convex side is of no importance. If, however, the 
column is made of a material, such as cast iron, whose tensile strength is much less 
than its compressive strength, the character and magnitude of the stress on the 
convex side should be investigated. If e' be the distance from the neutral 
axis to the outer fibre on the convex side, then the stress (p) on the convex side is, 

P r Pc f lY (o + a) e' "I Tr . . . , 

P = t I o ^ I — ) — 1 + -^ — —^ — i If P is positive the stress is ten- 

c A \__rn tt 2 E \ pj p 2 

sile; if p is negative the stress is compressive. 



STRAINING ACTIONS IN MACHINE ELEMENTS 75 

Assumed a diameter of 8". Then p = 2 and A = 50 

800 r 3,000 /?6o\ 2 10X4"! 

Whence p = 1 + ^ ,' (^— ) + * 

50 L 1 X - X 1,500,000 \ 2 / 4 J 

4 

800 r "| 

= — 1 + 26 + 10 

5 o L J 

= 592 pounds per square inch. 

If this excess is considered too great, a second approximation 
can be made. 

22. Stress Due to Change of Temperature. Practically all 
metals expand when heated, and contract again when cooled. 
The amount which a bar expands per unit of length, for a rise 
of one degree in temperature, is called its coefficient of linear 
expansion, and will be denoted by C. The following table gives 
values of C for various substances for one degree Fahrenheit : 

Hard Steel C = .0000074 

Soft Steel .C= .0000065 

Cast Iron C = .0000062 

Wrought Iron C = .0000068 

If a bar of metal is held at the ends, so as to prevent it from 
expanding or contracting, stresses are produced in it which are 
called temperature stresses ; the effect being the same as though 
the bar had been compressed, or elongated, an amount corre- 
sponding to its expansion or contraction due to the change in 
temperature. 

Let / = change in temperature in degrees. 

Let/> = stress induced per unit area. 

stress p 

Since E = — = —■ . . p = CtE 

strain Lt 

Example. A bar of wrought iron 2" square is raised to a 
temperature of 100 degrees above its normal. If held so that it 
cannot expand, what stress will be induced in it, and what force 
must oppose it to prevent expansion ? 



76 MACHINE DESIGN 

Let E = 30,000,000 

P = C t E = .0000068 X 100 X 30,000,000 = 20,400 lbs. 
and the total opposing force P will be 

P = 20,400 X 4 = 81,600 lbs. 

23. Resilience. In all the previous discussions on the vari- 
ous straining actions to which a member may be subjected, it 
has been assumed that the load was a simple dead load and 
applied without initial velocity or impulse. But, as already 
pointed out, the load may be applied impulsively; or it may be 
applied in any way, and removed and applied again and again 
repeatedly. The application of a load in an impulsive manner, 
or the repeated application of a load, does not affect the charac- 
ter of the straining action, but does affect the amount of stress or 
strain. In order to more clearly discuss the effect of impulsive 
loading it will be necessary to consider the straining effect of a 
load somewhat more fully; the discussion of repeated loads will 
be given in a succeeding section. 

If a material is distorted by a straining action, it is capable of 
doing a certain amount of work as it recovers its original form. 
If the deformation does not exceed the elastic strain, this amount 
of work is equal to the work done upon the material in producing 
such deformation. If the material is strained beyond the elastic 
limit, it returns work only equal to that expended in producing 
elastic deformation; and the energy required to cause the plastic 
deformation, or set, is not recovered, as it is not stored but has 
been expended in producing such permanent change of form. 
Ordinary springs illustrate the first case; the Shaping of ductile 
metals by forging, rolling, wire-drawing, etc., are processes in 
which nearly all of the energy is expended in producing perma- 
nent deformation. 

The work required to produce a strain in a member is called 
the work of deformation. If the strain produced is equal to 
the deformation at the true elastic limit, the energy expended is 
called elastic resilience* II the piece is ruptured, the energy 

* When the term resilience is used without qualifying context, elastic resitteikce. 
is to be understood. 



STRAINING ACTIONS IN MACHINE ELEMENTS 77 

expended in breaking it is called total work of deformation. 

If O ad e (Fig. 6) is the stress-strain diagram for a given mate 
rial, the area O a a' represents the elastic resilience, and O a d e e' 
represents the total work of deformation per cubic inch of tlw 
material. 

In such materials as have well-marked elastic limits (propor- 
tionality between stress and strain through a definite range) the 
line Oa is a sensibly straight line, and the elastic resilience 
Oaa' = %aa' X Oa'; or, the elastic resilience equals the elastic strain 
{Oa') multiplied by one-half the elastic stress (X aa'). The area 
Oadee' equals the base (Oe') multiplied by the mean ordinate (y) 
of the curve Oade; or, if the quotient of this mean ordinate of 
the curve divided by the maximum ordinate be called k, the 
work of deformation equals the ultimate strain multiplied by k 
times the maximum stress. It is evident that for a straining 
action beyond the elastic limit, k > % and k < t. 

The curve OADEE' represents the stress-strain diagram of a 
material having higher elastic and ultimate strength than the 
former. The greater inclination of the elastic line (OA) with 
the axis of strain (OX) shows, in the second case, a higher modulus 
of elasticity, as this modulus equals the elastic stress divided 

a a' 
by the elastic strain. In the first case E, = -rr—. in the sec- 

Oa 

A A' 
ond case, E 2 = _ .. . 
' 2 O A' 

The stress-strain diagram OADEE' shows that of two mate- 
rials one may have both the higher elastic and ultimate strength, 
and still have less elastic resilience and less total work of deforma- 
tion. If the curve O a" d" e" is the stress-strain diagram of a 
third material (having a modulus of elasticity similar to the 
first), it appears that this third material possesses greater elastic 
resilience, but less total work of deformation than the first. 

A comparison of these illustrative stress-strain diagrams (for 
quite different materials) also shows that, for a given stress, the 
more ductile, less rigid material may have the greater resilience. 
Hence, when a member must absorb considerable energy, as in 



78 MACHINE DESIGN 

case of severe shock, a comparatively weak yielding material may 
be safer than a stronger, stiffer material. This is frequently 
recognized in drawing specifications. The principle is similar to 
that involved in the use of springs to avoid undue stress from 
shock. In fact springs differ from the so-called rigid members 
only in the degree of distortions under loads, or in having much 
greater resilience for a given maximum load. 

If a material is strained beyond its elastic limit, as to a' (Fig. 
13 b), upon removal of the load it will be found to have such a 
permanent set as O O' '. Upon again applying load, its elastic 
curve will be O f a'; but beyond the point a' its stress-strain dia- 
gram will fall in with the curve which would have been pro- 
duced by continuing the first test (i.e., a! de). Similarly, if 
loaded to a", the permanent set is O O", and upon again apply- 
ing load, the stress-strain diagram becomes O" a" d e. The 
elastic limit a" of the overstrained material is evidently higher 
than the original elastic limit, a; while the original total work of 
deformation, O a d e, is considerably greater than the total work 
of deformation of the overstrained material, O" a" d e. The 
effects of strain beyond the elastic limit are thus seen to be : 

I. Elevation of the elastic strength and increase of the elastic 
resilience. 

II. Reduction of the total work of deformation. 

These facts have an important influence on resistance to re- 
peated shock. The above noted elevation of the elastic limit by 
overstraining can usually be largely or wholly removed by 
annealing. 

24. Suddenly applied Load, Impact, Shock. It will perhaps 
be well to first consider the general case of a load impinging on 
the member, with an initial velocity; this velocity (v) correspond- 
ing to a free fall through the height h. For simplicity, the dis- 
cussion will be confined to a load producing a tensile stress; but 
the formulas will apply equally well to uniform compressive and 
shearing stresses, and all except (5) apply directly to cases of 
torsion and flexure. 



STRAINING ACTIONS IN MACHINE ELEMENTS 



79 



W = static value of load applied to member. 

h = height corresponding to velocity with which load is 
applied. 

<5 = total distortion of member due to impulsive load. 

p = maximum intensity of resulting stress. 
A = area of cross-section of the member. 
P = p A= total max. stress due to load as applied suddenly. 

A = total distortion of member due to static load, W. 
x = h + l (for convenience). 

k = & constant; its value is % if E. L. is not passed; but if 

E. L. is exceeded k > ** 



i and k < i. 




Fig. 13 (c). 



Fig. 13 (b). 



The energy to be absorbed by the member due to the impulsive 
application of the load is W (h + $); the work of deformation is 
k Pd. (See preceding article, Resilience.) 

Case I. — Maximum Stress within Elastic Limit. 

W (h + i) = kP3 =K P8 (1) 

PX 



d: I ::P:W .' . d = 



W 



_ 2Wh nr 2W 2 h 

P - — ^— + 2W = -yy + 2W 



(2) 
(3) 



80 MACHINE DESIGN 

2 IP/* 
X 



.'.P 2 =— T- + 2WP .'. P = W(l + Vl + 2x) . (4) 



p w 

P = -J=^^+Vl + 2X) (5) 

p x 

8 =— = X(i + a/ 1 + 2x) (6) 

The elongation at the elastic limit equals F + E, in which E = 
modulus of elasticity and F = intensity of stress at the elastic 
limit. 

If L = length of the member, 

(A ^ L) : (F ^ E) : - (W -^ A) :F; .'. X =WL + A E. . (7) 
As X is small for metals (except in the forms of springs) a 
moderate impinging velocity may produce very severe stress. It 
will be evident that X and 8 are directly proportional to the length 
of the' member; hence the stress produced by a given velocity of 
impact (height h) is reduced by using as long a member as pos- 
sible. 

If the load is applied instantaneously, but without initial veloc-» 
ity, h = o and x = o; whence 

p = w (1 + vTTo) = 2W ( 4 o 

p 2 w 
t = -Z = -A__ ; • • «) 

« = l(i+v / i + o)=2l . . . . . . . (60 

Case II. — Maximum Stress Beyond the Elastic Limit. If the 
maximum stress exceeds the elastic limit, the constant k of equa- 
tion (1) is between % and 1 (see Art. 23, Resilience), and its exact 
value cannot be determined in the absence of the stress-strain dia- 
gram for the particular material. Thus (Fig. 13 c), W (h + $), 
is represented by the rectangle mncq; and this area must equal 
the area O a b c; the latter being greater than the elastic resili 
ence, O a a', and less than the total work of deformation O ade e' f 
in this illustration. 

When the stress-strain diagram is known, the following prob- 
lems can be readily solved : — 

(a) Determination of the velocity of impinging of a given load 
(or corresponding value of h) to produce a given stress, or strain. 



STRAINING ACTIONS IN MACHINE ELEMENTS 8 1 

(b) Determination of the load which will produce any par- 
ticular stress, or strain, when impinging with a given velocity. 

(c) Determination of the stress, or strain, produced by a given 
load impinging with a given velocity. 

Let the work of deformation corresponding to the known 
stress, or strain, in (a) and (b), be called R = k P d. If the 
stress-strain diagram is for stress per unit of sectional area and 
strain per unit of length of the member, let W be the load per 
unit of sectional area; W the height due the velocity of impinging 
divided by the total acting length of the member; <*' the distor- 
tion per unit of length of the member due to impulsive load ; and 
R f the resilience for unit of volume, or the modulus of resilience. 

(a) : W Qi' + if) = k p V = R'. . ' . h f = — — <*' . . . (7) 

W 

(b) : W = 77^—7 (8) 

(c) : The solution of this problem is not quite so definite, in 
the general case, as the preceding; but it can be easily accom- 
plished, graphically, with sufficient accuracy. Draw the line g q 
(Fig. 13 c) (indefinitely), parallel to O e', and at a distance from 
it equal to W\ take out the area fig = gOt. Whatever the 
value of y, the shaded area O c qfi g O = W d f ; hence the un- 
shaded area under the stress-strain curve must equal W h\ A 
few trials will suffice to locate the limiting line b q c which will 
give/ ib qf = mnO t = W h'. 

The case in which the maximum stress is within the elastic 
limit is by far the most important, as it is almost always desired 
to keep the maximum intensity of stress, P + A, within the 
elastic limit, especially as every overstrain (beyond this limit) 
raises the elastic limit and decreases the total resilience (see Fig. 
13). The effect of a shock which strains a member beyond the 
elastic limit is to reduce its margin of safety for subsequent 
similar loads, because of reduction in its ultimate resilience. 
Numerous successive reductions of the total resilience by such 
actions may finally cause the member to break under a load 
which it has often previously sustained. 
6 



82 MACHINE DESIGN 

No doubt many cases of failure can be accounted for by the 
effects just discussed; but there is another and quite different 
kind of deterioration of material, which is treated in the follow- 
ing article. 

Dr. Thurston has shown that the prolonged application of a 
dead load may produce rupture, in time, with an intensity of 
stress considerably below the ordinary static ultimate strength 
but above the elastic stress. It is well known that an apprecia- 
ble time is necessary for a ductile metal to flow, as it does flow 
when its section is changed under stress; hence, a test piece will 
show greater apparent strength by quickly applying the load 
than by applying it more slowly, provided the application of 
load is not so rapid as to become impulsive. 

The kind of failure which is the subject of the next topic is due 
to a real permanent deterioration of the metal, and it is due to 
distinctly different causes from those mentioned above. 

25. On the Peculiar Action of Live Load. Fatigue of Metals. 
It has been found by experience and experiment, that materials 
which are subjected to continuous variation of load cannot be 
depended upon to resist as great stress as they will carry if applied 
but once, or only a few times. When the load is suddenly applied, 
and frequently repeated, the decline of strength or of the power of 
endurance may perhaps be ascribed, in part at least, to the eleva- 
tion of the elastic limit and reduction of the ultimate resilience, as 
discussed in Art. 24. But apart from this cause, with repeated 
loads, even in the absence of appreciable shock, a decided de- 
terioration of the material very frequently occurs. This effect 
has been called the Fatigue of Materials, although some authorities 
restrict this term to the kind of deterioration already referred to as 
the simple result of a decrease of resilience. The term fatigue 
implies a weakening of the material due to a general change of 
structure. It was formerly supposed that the repeated variation 
of stress caused such change of the general structure, possibly 
owing to slight departure from perfect elasticity under stress 
much below that ordinarily designated as the elastic limit. The 
crystalline appearance of the fracture sustained this view; but 
numerous tests of pieces from a member ruptured in this way. 



STRAINING ACTIONS IN MACHINE ELEMENTS 8$ 

(taken as near as possible to the break) , fail to show such crystalline 
fracture, and it is difficult to reconcile the normal appearance and 
behavior of such test pieces with the theory of general change of 
structure. 

A theory which has been largely accepted is that every piece 
of metal contains innumerable minute flaws or imperfections, 
often originally too small to be detected by ordinary means. These 
"micro-flaws" tend to extend across the section under variation 
of stress, and may, in time, reduce the net sound section so greatly 
that the intensity of stress in the fibres which remain intact be- 
comes equal to the normal breaking strength of the material. 
Professor Johnson suggests: "the gradual fracture of metals" as 
a more appropriate term than " fatigue." Many men of large prac- 
tical experience still prefer wrought iron to mild steel for various 
members which are subject to constantly reversing stress. 

It is probable that the prejudice against steel is largely the 
result of unskilful manipulation of this more sensitive mate- 
rial ; and the product of the best steel makers of to-day is much 
stronger and more reliable than wrought iron. 

However, it is just possible that the very lack of homogeneity 
in wrought iron renders it safer under varying stress (other things 
being equal) , as the fibres are more or less separated by the streaks 
of slag, and a flaw is less apt to extend across the entire section 
than it is in the continuous structure of steel. Wrought iron may 
be likened to a wire rope, in which a fracture in one wire does 
not directly extend to adjacent wires. 

The " gradual fracture" through extension of " micro-flaws" 
seems to accord with the observed facts more closely than the 
older theory of general change of structure. 

In the American Machinist (Sept. 27, 1906) will be found an 
account of recent researches tending to show that metals are 
made up of grains, each grain consisting of many crystals, and 
that when deformation takes place in a metal these crystals move 
relatively to each other along " gliding planes.'' 1 If the stress 
producing such sliding is repeated often enough the contact at 
the gliding planes weakens and finally passes into a crack or 
series of cracks which extend across the section. 



84 MACHINE DESIGN 

The theory of the subject is, as yet, too incomplete to permit 
of derivation of rational formulae to account for the effects of re- 
peated live loads; and if the " micro-flaw " theory is correct, it 
is not probable that such rational analysis can ever be satisfac- 
torily applied. 

All of the formulae that have been derived for computation of 
breaking strength under known variations of load, or stress, are 
empirical ones which have been adjusted to fit the experiment- 
ally determined facts. 

Consult: Johnson's " Materials of Construction." 
Merriman's " Mechanics of Materials." 
Unwin's "Testing of Materials." 
Weyrauch (Du Bois) : " Structure of Iron and 
Steel." 

Experiment has shown that the breaking strength under re- 
peated loading, or the " carrying strength," is a function of the 
magnitude of the variation of stress and of the number of repeti- 
tions of such varying stress. Furthermore, this function is 
different for different materials; and there are authentic observa- 
tions on record which go to show that, as between different mate- 
rials, the one with the higher static breaking strength does not 
always possess the greater endurance under repeated loading. In 
general, however, the carrying strength under repeated loads is a 
function of the static strength. 

The allowable working stress usually depends upon : (a) The 
number of applications of the load. This should be considered 
as indefinite, or practically infinite, in many machine members, 
(b) The range of load. This is frequently either from zero to 
a maximum; or between equal plus and minus values, (c) The 
static breaking strength or the elastic strength. 

The first systematic experiments upon the effect of repeated 
loading were conducted by Wohler [1859 t0 1870]. He found, 
for example, that a bar of wrought iron, subjected to ten- 
sile stress varying from zero to the maximum, was ruptured 
by: 



STRAINING ACTIONS IN MACHINE ELEMENTS 



85 



800 repetitions from o to 52,800 lbs. per sq. in. 
107,000 " " o to 48,000 " " 

450,000 " " o to 39,000 

10,140,000 " o to 35,000 " 

— Merriman, page 191. 
It was found that the stress could be varied from zero up to 
something less than the elastic limit an indefinite number of 
times (several millions) before rupture occurred; but with com- 
plete reversal of stress, or alternate equal and opposite stresses, 
(tension and compression), it could be broken, by a sufficient 




Fig. 13 (d). 

number of applications, when the maximum stress was only 
about one-half to two-thirds the stress at the elastic limit. 

A number of efforts have been made to deduce from the exper- 
iments of Wohler, formulae which could be applied to the design 
of machine members (see Unwin, page 36). One of the best of 
these formulae is that of Professor Johnson as it is easily applied 
to all cases that will arise; it is simpler than most of those previ- 
ously proposed; and it is probably as reliable as any yet offered. 

Two formulae which have been very generally accepted for 
computing the probable carrying strength are: Launhardt's for 



86 MACHINE DESIGN 

varying stress of one kind only, and Weyrauch's for stress which 
changes sign. 

Suppose a material to have a static ultimate strength u of 
60,000 lbs. per sq. in. If the minimum unit strength be plotted 
as a straight line, A O B (Fig. 13 d), the locus of the maximum 
unit stress, from the Launhardt formula, is the broken curve 
from B to D. That is, for example, when the minimum tensile 
stress is 12,500, the maximum tensile carrying stress would be 
about 40,000; or the material could be expected to stand an 
indefinite number of loadings if the range of stress did not 
exceed 15,000 to 40,000 pounds per square inch in tension. In a 
similar way, the broken curve from D to C is the locus of maxi- 
mum tension, from the Weyrauch formula, when the locus of 
minimum stress (negative tension, or compression) is the straight 
line A O. It will appear that the straight line C D B agrees 
fairly well with these two curves. Inasmuch as it seems un- 
reasonable to expect an abrupt change of law when the minimum 
stress passes through zero, and as there is no rational basis for 
the Launhardt and Weyrauch formulae, it appears reasonable to 
adopt the upper straight line as the locus of the maximum stress. 
Owing to the discrepancies in the observations (which must be 
expected from the probable cause of the deterioration of the 
metal), this straight line may be accepted as representing the 
law as accurately as could be expected of any empirical line. 
These are, in substance, the reasons given by Professor Johnson 
for basing his formula on the straight line C D B. For full dis- 
cussion and derivation of the following formula, see Johnson's 
"Materials of Construction," pages 545-547. 

Let p 2 = maximum intensity of stress. 
p x = minimum intensity of stress. 
u = ultimate (static) intensity of stress. 

Then in general : 

*- — 7a (I) 

h 

As the expressions contain the ratio of the minimum to maxi- 
mum intensities of stress, instead of their difference, they are ap- 



STRAINING ACTIONS IN MACHINE ELEMENTS 87 

plicable when the area of cross-section of the member is unknown; 
for whatever this area, the ratio of the stresses is the same as the 
ratio of the loads producing these stresses. In substituting values 
of p x and p 2 , care must be taken to use proper signs; thus, if 
tension is taken as positive, compression is negative; or, if the 
stress varies between tension and compression p 2 is positive and 
p x is negative. 

For dead load, p x = p 2 ; 

>''P2= V =Z ~^ = U ,....' (2) 

1 — y 2 h A 

A 

P 

For repeated load when p 1 = o, — = o 

P2 

• • A = tzi^ = /2 u (3) 

For complete reversal of load, p x = — p 2 

P2 = Z7^~ = T , 1/ = A u (4) 

'2 



I + L 



+ P2 

The three special cases (2), (3), and (4), are those most com- 
monly met with in designing, but the general expression (1) 
should not be lost sight of. 

Example. A bar of steel, whose ultimate static tensile strength 
is 70,000 lbs. per sq. inch, is subjected to a repeated load whose 
minimum value is one half the maximum value. What is the 
maximum stress that can be carried by the bar for an indefinite 
number of repetitions? 

h 



Since the stress will be proportional to the load p x = 



2 



1 . . " . A x 70,000 

Hence substituting in equation (1), p 2 = — = 47,000. 



2 2 p 2 
It is to be noted that the allowable maximum stress is above the 
original elastic limit of most steel, and if the piece were designed 
to be stressed to 47,000 lbs. the result would be that the first 
application of the load would raise the elastic limit to that value. 



88 MACHINE DESIGN 

But the piece would take permanent set and be in most cases of 
no further use. A factor of safety must therefore be used in 
order that the maximum stress may be well below the elastic 
limit. 

The experiments of Wohler, and his successor in the field, 
Baushinger, were conducted on a very limited variety of mate- 
rials; so that while the above discussion points out what may be 
expected in a general way from most materials, they are not suf- 
ficiently conclusive to make it possible to pick out the exact 
factor of safety to be used in all cases. They do, however, throw 
much light on the apparently high factors of safety which must 
sometimes be used, and for which no other satisfactory explana- 
tion has been found. 

26. The Factor of Safety. The preceding paragraphs (arti- 
cles 9 to 26) have considered the effect that different methods 
of applying the load will have on a member, and the relations 
which exist between a given dead load and the resulting stress 
and strain. It has been shown in Art. 24 that if the load is 
applied suddenly the resulting stress and strain will be twice as 
great as for a dead load. And finally in Art. 25 it has been 
shown that the maximum stress that can with safety be induced 
repeatedly in a member, will depend on the range of stress. It 
would seem as though a member designed in accordance with 
these logical theories would be satisfactory. But it must be 
remembered that these theories are not absolute, that the in- 
formation regarding the characteristics of materials is still very 
incomplete; that flaws and hidden defects always exist; and 
finally that there is always danger of accidental overloading. 

In addition, it is generally essential that a machine member be 
not only strong enough to avoid breaking under the regular 
maximum working load, but also that it shall not receive a 
permanent set ; for a machine member ordinarily becomes useless 
if it takes such set after it has been given the required form. In 
many cases a temporary strain, even considerably below that 
corresponding to the elastic limit, would seriously impair the 
accuracy of operation; and in such cases the member often re- 
quires great excess of strength to secure sufficient rigidity. It 



STRAINING ACTIONS IN MACHINE ELEMENTS 89 

follows, therefore, from these considerations that if the design of 
a machine member were based on the maximum allowable stress, 
as indicated by Wohler's experiments (such stress being modi- 
fied by the theory of suddenly applied loading, should it be 
present), there would be no margin to allow for the uncer- 
tainties and unknown defects enumerated above; and in many 
cases leave no assurance that the elastic limit would not be ex- 
ceeded. So that while stresses fixed in accordance with these 
theories form a good basis, they must in general be reduced by 
means of a factor of safety so that the working stress is enough 
lower to provide for these uncertainties. 

The factor of safety is generally defined as the quotient of the 
ultimate static strength divided by the working stress. A con- 
sideration of Wohler's experiments shows that such a definition 
is misleading. For a factor of safety of 2, for instance, might 
be perfectly safe for a dead load; but for a repeated load with 
stress in one direction it would leave no margin at all for contin- 
gencies. The apparent factor of safety would seem to be a 
better term, and the real factor of safety may be defined as the 
quotient of the carrying strength, or maximum allowable stress 
as given by Wohler's experiments, divided by the working stress. 

The factor of safety has been called the "factor of ignorance," 
and, as it is too often applied, it is perhaps little else. Thus 
very often it is specified that all the members of a machine shall 
be designed with a certain fixed factor of safety without regard 
to the conditions under which the various members may have to 
act. A factor of safety applied in this manner is, generally 
speaking, a factor of ignorance. It is probable that the factor 
of safety will always retain an element of ignorance, for it can 
hardly be hoped that the powers of analysis will ever permit the 
prediction of the exact effect of every possible straining action, 
due to regular service and accident. Neither can it be expected 
that the methods of manufacture, and inspection, will become so 
perfect as to eliminate or measure precisely every possible defect 
in materials or workmanship. But a careful study of the condi- 
tions of each particular case and a proper attention to the effects 
which may be weighed (at least approximately) should, with 



90 MACHINE DESIGN 

the knowledge now to be had, enable the designer to make a 
fairly accurate application of the factor of safety, an intelligent 
choice of which is the most important part of design. 

Most of the formulae of Mechanics which are applicable to the 
design of machine members, are based on theoretical treatment 
of the stresses induced by the action of given forces within the 
elastic limit upon the member under consideration; and the 
theoretical conclusions so reached are amply verified by practical 
experiment. When, therefore, the conditions under which the 
member is to work can be analyzed, and the laws of Mechanics 
applied to its design, such methods as outlined in this chapter 
are perfectly rational, if intelligent allowance is made for contin- 
gencies. Many machine members, however, are subjected to 
such a complicated system of stress that analysis cannot be 
strictly applied, and less satisfactory approximations or assump- 
tions are unavoidable in the present state of knowledge. When 
such is the case, the designer must either base the design on the 
predominating stress, if there is such, allowing such a margin or 
factor of safety as experience or experiment may show, to pro- 
vide for the minor uncertain stresses; or, if the case considered 
be beyond such treatment, recourse must be had to empirical 
methods or judgment. (See Art. i.) 

While therefore mathematical treatment of any case will serve 
as a good guide to correct proportions, such treatment must 
always be tempered with judgment, a high development of 
which is necessary to successful design, as in all other branches 
of engineering. 

While, also, no fixed rules for selecting the factor of 
safety can be laid down, a knowledge of Wohler's experiments, 
and the effect of suddenly applied loads, will greatly aid the 
designer in the matter. Thus when it is known that the load is 
to be a dead load, an apparent factor of safety of 3 will, for 
wrought iron, or steel, bring the working stress well below the 
elastic limit and allow something for contingencies. If, however, 
the load be a repeated load, the stress varying from zero to a 
maximum tensile stress, the apparent factor of safety for steel 
must at least be 5, to allow a good margin below the elastic limit £ 



STRAINING ACTIONS IN MACHINE ELEMENTS 



91 



and in either case, if in addition the load is to be suddenly applied, 
these factors must be multiplied by 2 to insure safety. 

Example. A steel beam is subjected to a suddenly applied 
load which alternately induces an equal tensile and compressive 
stress; if the ultimate strength be 60,000 lbs. per sq. in., what 
apparent factor of safety should be used, and what will be the 
real factor of safety? 

Since the stress is a reversed one, the maximum allowable 
stress or carrying strength is by Wohler's experiments one-third 
of ultimate strength or 20,000 lbs. If the working stress is one- 
half of this value or 10,000 lbs., it will leave a good margin for 
contingencies, disregarding the impulsive effect. But the load 
is applied suddenly and, by Art. 24, this value (10,000) must 
be again divided by 2, making the working stress 5,000 lbs. per 

60,000 
sq. in. Therefore the apparent factor of safety is — - = 12, 



5,000 



while the real factor based on Wohler's law is 



20,000 
5,000 



= 4. 



If the member should have to work under extremely trying 
conditions, or if shock or other stresses which could not be 
analyzed were present, this value might have to be still further 
reduced 

TABLE IV 

FACTORS OF SAFETY 





Dead 
Load. 


Repeated 
Stress in One Direction. 


Repeated 
Reversed Stress. 


Character of 
Material. 


Gradually 

Applied 

Load. 


Suddenly 

Applied 

Load. 


Gradually 

Applied 

Load. 


Suddenly 

Applied 

Load. 


Wrought Iron, Steel, 
or other Ductile 
Metals 

Cast Iron, or other 
Brittle Metals 


3 
4 


« 
5 

6 


10 
12 


6 
10 


12 
20 



Table 4 contains factors of safety such as are used in practice 
and which agree fairly well with the foregoing theory. They 
are, of course, average values and must be used with judgment; 



92 MACHINE DESIGN 

but in the absence of trained judgment, or as an aid to its de- 
velopment, they may be found useful. 

Table 5 contains values of the ultimate strengths and elastic 
limits of the materials most used in engineering. They, also, are 
average values such as the designer must use in the absence of 
exact information regarding the material to be employed, and in 
general such exact information is lacking. 

It may be observed that an increased factor of safety may not 
always in the case of cast metals give a stronger member. If 
the increased dimensions give sections so thick that sponginess 
results, the gain in strength may be negative; and when internal 
pressure, such las is found in hydraulic work, is to be withstood, 
it is often necessary to do with a smaller factor of safety to 
insure soundness. 



STRAINING ACTIONS IN MACHINE ELEMENTS 



93 



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94 



MACHINE DESIGN 
TABLE VI 





Character of Stress 
or Strain. 


Formula. 


A 




P 


B 


Strain in Ten. or Comp 


A- Pl 

AE 


c 




P 






D 


Torsional Stress 


Pa _ r _ P' T p 






e 


E 


Torsional Stress, Solid Cir- 
cular Shaft 


_ £>. it d 3 
Pa = T = f' 

10 








F 


Torsional Stress, Hollow Cir- 
cular Shaft 


10 d\ 








G 


Torsional Strain, Solid Cir- 
cular Shaft 


0= * 2Tl 

7T E 8 d* 








H 


Torsional Strain, Hollow Cir- 
cular Shaft 


e- 32Tl 




tt E s (d!* — d 2 4 ) 








I 




See Table I. 


J 


Stress due to Flexure 


M = tL See Table I. 
e 




Combined Bend'g and Twist'g 




K 


M e =y 2 M-\-y 2 v m 2 + 7* 




a tt u a 




K, 


T e = V M 2 -\-T 2 


K 2 


n a it u 


M e = %[x+V * a +i]r 




u a (I a 




K 3 


r e = rV^ + i 




Combined Torsion and Com- 
pression 




L 


t 2 \p\ jp* 1 64r n 




Pc ~ ird 2 L V ' ^ J 








L, 


Combined Torsion and Com- 
pression 


2 A / o 64 T 2 








M 


Combined Flexure and Direct 
Stress 


P . Pae 








N 


Long Column. 


A' P > 






m7r\E\p) 


O 


Eccentric Loading of Long 
Columns 


PT /> c //\ 2 (a + «)'"] 




' ^ f m 7r 2 £V/oJ ' X> 2 









STRAINING ACTIONS IN MACHINE ELEMENTS 



95 



TABLE VII 

PROPERTIES OF SECTIONS 



Shape 

of 
Section. 



Moment 

of 
Inertia. 

I 



Modulus 

of 
Section. 

J_ 

e 



Square of 
Radius of 
Gyration. 
P 2 =l 



Polar 
Moment of. 
Inertia. 
J 



B*=.049d 4 
6» 



7TD 

~32~ 



= ..oaaD 3 



D 2 

.16 



ttD 4 
32 



f-V*" 



£M 



82 L D J 



•D^d 2 
16 



7T^D 4 -d 4 ) 

32 



h 



l«-B-*i 



BH' 
12 



BB 2 



12 



BH(B 8 +H 2 ) 
12 






I** 



\ reb-»i | 

*-B"»» 



i [BK s -bh 8 ] 



m™*-^ 



6h: 



j^ [ "BH 3 — bh 3 "| 
12 [bH — bh J 




i[BH 8 -b^] 



i[ BH ^bha 



6.H 



1 f ~BH 8 — bh 3 ~| 
12 |_BH bh J 




K--B-H 



^[BH 3 -bh 3 ] 



12 



s[BH 3 - bh 3] 



6H 



1 [ "BH 3 -bh 3 "] 
12|_BH— bh J 



IT * 

w/P/Jto 
; k-b» 
X'B-*! 




-— i?2 



1 = 



(BH 2 — bh 2 ) -4_BH bh (H-h) 2 
12(BH-bh) 



I^ (BH 2 -bh 2 )-4BHbh(H-h) 2 
ei = 6(BH 2 -bh 2 ) 

1 ^ (BH^bh 2 )— 4BHbh (H-h) 2 
e 2 6(BH 2 -2bhH+bh 2 ) 




Til>H 3 +Bh 3 ] 



K» hW ] 



6H 



bH'H-B'h^ 
12(bH+Bh) 



-*-* y- 

1 2H i» e, 
jl e 



K-B-*i 



;y 



f 2 - 



BH J 

36 



I. _,BH 2 

e x 24, 

I _BH 2 

e 2 12 



,Hf 
18 



hjl 



ttBH 3 

64. 



7TBH- 
32 



16 



tt(BH 3 +HB 8 



64 



!li. 



CHAPTER IV 

GENERAL THEORY OF FRICTION, LUBRICATION, AND 

EFFICIENCY 

27. Friction in General. When two solid surfaces are held in 
contact by any appreciable force, any effort tending to move them 
relatively to each other is met by a resisting force acting tangen- 
tially to the surface of separation of the two bodies. This resist- 
ance to relative motion is due to the interlocking of the minute de- 
pressions and elevations which exist even in the smoothest surfaces 
and will, of course, vary with different properties of materials 
and different qualities of finish. Thus, unsurfaced cast iron will 
show a very great resistance to relative motion, while two hard- 
ened and ground surfaces of steel will move over each other 
with much more ease. If the two surfaces are very carefully 
fitted together without any foreign matter in between, they will, 
in the case of many substances, adhere firmly together, which 
still further increases the resistance to relative motion. If oils 
or lubricants of any kind are interposed between the surfaces, 
the resistance to relative motion is, to a considerable extent, 
overcome. 

This tendency to resist relative motion is sometimes a desirable 
feature and sometimes not. In the case of bearing and rubbing 
surfaces generally, such frictional resistances result in loss of 
power and should be reduced to a minimum; while in the case 
of friction clutches, brake straps, keys, screw fastenings, etc., 
frictional resistance is of great utility and every effort is made 
to insure its presence. The laws of friction, and the manner 
of their application therefore, are of prime importance to the 
engineer. 

These laws are at present rather imperfectly understood, 
though considerable experimental work has been done. It 
has been found that many of the older theories based on ex- 

96 



GENERAL THEORY OF FRICTION 97 

perimental work are true only for the range of conditions 
covered by the experiments, and that conditions different from 
these show entirely different results. 

The ratio of frictional resistance F, to' the normal load P, is 
called the coefficient of friction; or if this ratio be denoted 

F 
by/, then for flat surfaces, f = — or F = fP. 

In the case of circular surfaces, such as journals and bearings, 
the distribution of the normal pressure is variable and depend- 
ent on the manner in which the surfaces are fitted together. 
In such cases it is customary for convenience to define the co- 
efficient of friction in a similar way as in flat surfaces, and con- 
sider that it has special values for circular surfaces. The co- 
efficient of friction for circular surfaces will be denoted by fj.. 
Hence as before, F = nP. 

The intensity of normal pressure on circular surfaces, as before 
stated, is difficult of accurate determination and it is therefore 
customary to take as the normal pressure the intensity of pres- 
sure per unit of projected area. Or, if d = diameter of shaft, 
and / = length of bearing, then the intensity of pressure per 

unit of projected area, w = — . 

a I 

The energy absorbed by frictional resistance is transformed 
into heat which is conducted away by conduction and radiation 
to the air, or, in the case of certain kinds of bearings, by water 
circulation or other means. The work of friction is often there- 
fore an important factor in the design of rubbing surfaces. For 
flat plates the foot pounds of energy absorbed per minute is 
E = fP V, where V the velocity is in feet per minute and P is 
in pounds. For circular surfaces, if N be the number of revolu- 
tions per minute, and d the diameter of shaft in inches, 

12 

* For other forms of surfaces see Kent's " Engineer's Pocketbook," page 938, 
and Thurston's "Friction and Lost Work," page 40. 
7 



98 MACHINE DESIGN 

If then /or t± be known, for any pair of rubbing surfaces, the 
frictional resistance and the energy absorbed for any load P may 
be calculated. Values of/ and /« have been obtained experimen- 
tally for many of the materials and conditions met with in engi- 
neering, but the data so far available are still incomplete. 

The consideration of the laws of friction, as applied to ma- 
chinery, naturally divides itself into two parts. 

(a) Friction of Dry or Unlubricated Surfaces. 

(b) Friction of Lubricated Surfaces. 

28. Friction of Unlubricated Surfaces. The experiments 
of Morin, Rennie, Coulomb* and many others, furnish the fol- 
lowing laws for dry or very slightly lubricated surfaces. 

(1) The frictional resistance is approximately proportional to 
the normal load. 

(2) The frictional resistance is approximately independent of 
the extent of the surfaces. 

(3) The frictional resistance, except at very low speeds, de- 
creases as the velocity increases. 

It was formerly supposed that an abrupt change took place in 
the value of / when the body passed from a state of motion to 
one of rest. It seems now, however, that while the coefficient 
of rest is in general greater than that of motion, the change in 
value is gradual and the value at rest is not far different from 
that at very slow motion. As the velocity increases, the value 
of / materially decreases and this must be taken account of 
in designing machinery where friction is involved. Unfortu- 
nately the information regarding high or even moderate speeds is 
also very incomplete. 

The following values of/ must, in view of the incomplete infor- 
mation, and also because of variations which come with slight 
changes of conditions, be looked on as approximate values only. 
Unless it is positively known that the surfaces will be kept free 
from even slight contamination by oily substances, these values 
must be, used with judgment. 

* See "Lubrication and Lubricants": Archbutt & Deely, for a full discussion 
of these points. 



GENERAL THEORY OF FRICTION 99 

Coefficients of Friction (/ ) for Dry or Slightly Lubricated 

Surfaces. 

Wood on Wood — Static or very low velocity 3 to .5 

Wood on Metals " " " " " 2 to .6 

Leather on Metals " " " il " 3 to .6 

Leather on Wood " " " " " 3*0.5 

Metal on Metal " " " " " (average) .3 

Cast Iron on Steel — velocity = 440 feet per minute. . . .32 

= 2640 " " " . . . .2 
5280 " " " . ■ . . .06 



a u a tt = 2640 u a 

11 11 a 11 u -«Q^ 11 11 11 



There are no experimental data giving the decrease in the 
value of/ at high speeds, for combinations such as wood or leather 
on metals. The data for cast iron on steel will, however, serve as 
a rough guide to what may be expected to occur. It is to be par- 
ticularly noted that, in designing brake shoes or other friction 
machinery where great velocities are involved, allowance must 
be made for the decrease in the value of the coefficient. 

29. Dry Rolling Friction. When a curved body rolls upon 
a plane or curved surface, it has been found that the so-called 
frictional resistance due to the rolling action is much less than 
that due to sliding, for the same load. If P = the load; F = the 
horizontal force required at the axis of a circular body to pro- 
duce and sustain uniform motion ; and r = radius of rolling body, 

kP 

it has been found that F = where k is a coefficient to be de- 

r 

termined experimentally. If r be expressed in inches k is found 
to have a value of about .02 for iron or steel rolling on iron or 
steel. 

Neither the coefficient k nor the exact theory of rolling friction 
is at present very accurately known. The most important use 
of rolling friction is, as far as the present discussion is concerned, 
in connection with roller bearings for shafting, and a fuller dis- 
cussion of these will be given later. 

30. Friction of Lubricated Surfaces. When a lubricant is 
interposed between a pair of rubbing surfaces, the frictional 
resistance is materially reduced because the surfaces are wholly or 



IOO MACHINE DESIGN 

partially separated from each other by the lubricant. The 
lubricant may be fed to the surfaces in a number of ways. If the 
motion is intermittent, and other conditions will allow, a simple 
oil hole leading to the rubbing surfaces is often used. If the 
motion is continuous, some form of oil cup which will give a con- 
tinuous supply is better. Fig. 14 (a) shows a cup of the 
simpler type w T here a wick, of cotton or wool draws up the oil by 
capillary attraction and feeds it slowly into the oil hole. This 
is sometimes called siphon feed. Fig. 14 (b) shows a so- 
called sight feed cup where the oil falling by gravity from the 
cup can be seen as it passes the hole e and the flow can be regu- 
lated by the screw d. Centrifugal action is also used to some 
extent to feed oil to rotating parts. Sometimes an opening is 
made in the bearing so that a pad saturated with lubricant can be 
kept pressed up against the moving surface, thus lubricating the 
whole length of the journal continuously. For heavy lubricants, 
such as greases, where very heavy pressures are carried on the 
rubbing surfaces, so-called compression cups are often used and 
are constructed so as to force the lubricant in between the sur- 
faces. Fig. 14 (c) shows a "ring oiled" bearing. The ring r 
running loose on the shaft s dips into the pocket below the shaft. 
The friction of the ring on the shaft causes it to rotate and draw 
up oil from the pocket. Sometimes chains are used instead of 
solid rings. For the most efficient lubrication the journal itself 
runs in a bath of oil (Fig. 1 5) or is flooded with oil supplied under 
pressure. The relative merits of these various methods of sup- 
plying the lubricant will be more apparent after a discussion of 
the general laws of lubrication. 

The effect of friction, and the efficiency of lubrication of so- 
called lubricated surfaces, may conveniently be treated under 
three heads :— 

(a) Static Friction of Lubricated Surfaces. 

(b) Friction of Imperfectly Lubricated Surfaces. 

(c) Friction of Perfectly Lubricated Surfaces. 

31. Static Friction and Lubrication. When a pair of lu- 
bricated surfaces are pressed together by a load, the pressure 
tends to slowly expel the lubricant from between the surfaces. 



GENERAL THEORY OF FRICTION 



IOt 



Experiments and experience show that it is very difficult even 
with limited areas and heavy pressures completely to expel the 
lubricant. If ordinary machinery, however, is allowed to stand 
at rest for a short period of time, this action is sufficient to expel 
so much of the lubricant that may have been between the sur- 
faces while running as to allow the metallic surfaces to come 
more or less in contact. The static coefficient of friction of lubri- 
cated surfaces is hence very much higher than that of surfaces 
which move even very slowly; for it will be seen presently that even 
at low velocities the surfaces tend to draw in the lubricant by their 
motion. It is a well-known fact that heavy machinery always 
offers a great resistance to starting after lying idle a short time 
and often the rubbing surfaces, if not oiled before starting, will 




Fig. 15. 



abrade each other before the lubricating action due to running 
begins to take effect. The materials therefore for the rubbing 
surfaces of heavy machinery should be carefully chosen for their 
antifriction qualities, and oil grooves should be carefully provided 
so that lubricant can be applied as near the point of greatest 
pressure as possible before motion begins. 

The coefficient of static friction for lubricated surfaces is not 
very accurately known and it varies somewhat with the pres- 
sure and character of the lubricant. A fair average value for 
metal surfaces and pressures ranging from 75 to 500 lbs. per sq. 
in. is .15.* 



*See Thurston's "Friction and Lost Work," pages 316-317. 



102 MACHINE DESIGN 

32. Imperfect Lubrication. When one lubricated surface 
slides over another, the moving surface, even at low velocities, 
tends to carry the lubricant, if properly applied, in between the 
surfaces. Thus the layer of oil which touches the surface of a 
journal adheres to it and is carried along under the bearing. 
This layer in turn tends to carry along the layer which next ad- 
joins it, because the viscosity of the lubricant opposes the shear- 
ing action which results between layers on account of the action 
of the moving surface of the journal. In plane sliding surfaces 
the lubricant is generally applied to the stationary surface and 
tends to cling to it in spite of the tendency of the slider to rub 
it off. The action of the sliding surfaces in d-rawing in the lu- 
bricant is similar to that of the rotating journal, but in a much 
less marked degree as would naturally be expected from the na- 
ture of the case. If the velocity of rubbing be very low, or the 
pressure very high, or the supply of lubricant limited, the quantity 
of lubricant that is carried in is very small and the surfaces in 
contact are very slightly lubricated and may even be in actual 
metallic contact. The materials, therefore, for the rubbing sur- 
faces of slow-moving machinery should also be carefully chosen 
for their antifriction qualities, as even after the machinery has 
been successfully set in motion metallic contact may occur 
between them. 

If the velocity of rubbing and the supply of lubricant be in- 
creased, the load remaining the same, more and more lubricant 
is thrust between the surfaces by the action noted above till, at 
a point depending on the pressure, velocity of rubbing, and vis- 
cosity of the lubricant, the metallic surfaces are completely sep- 
arated and the friction becomes only that due to the fluid fric- 
tion of the lubricant itself. This last state is known as perfect 
lubrication. The formation of this separating film with in- 
creasing speed is probably gradual and the character of the 
contact most probably passes through a gradual change, from 
contact which is nearly metallic through successive stages of par- 
tially fluid contact to complete fluid separation. The exact 
point at which perfect lubrication occurs for any given load, 
velocity, and lubricant is not accurately known, but what data are 



GENERAL THEORY OF FRICTION 103 

available will be given in connection with the discussion of per- 
fect lubrication which follows. It is known, however, that per- 
fect lubrication cannot be obtained without a plentiful supply of 
the lubricant, as in the case where a journal runs in an oil bath, 
or is supplied by so-called forced lubrication where the lubricant 
is delivered under pressure. It is impossible or inconvenient, 
however, to lubricate the greater part of the rubbing surfaces of 
machines in this manner and, therefore, all surfaces lubricated 
by such means as simple oil holes, oil cups, oily pads, etc:, 
where the supply of lubricant is in any way restricted must be 
considered as imperfectly lubricated. 

As already noted, the exact condition which will exist between 
such surfaces depends on the pressure, the velocity of rubbing, 
the supply and character of the lubricant, and the temperature of 
the bearing as affecting the viscosity of the oil. Naturally where 
so many variables exist, experimental results are very discord- 
ant, and while an immense amount of work has been done, the 
results only serve to emphasize the great variation in conditions 
with change of these variables. It is evident, for instance, that 
if velocity and pressure remain constant, almost any condition 
may be produced from metallic contact to perfect lubrication 
simply by varying the supply of lubricant. The law of varia- 
tion of the coefficient of friction, with either varying pressure 
or velocity, is also found to be modified by the rate at which 
oil is supplied. The generally accepted theories for imperfectly 
lubricated bearings running under average conditions, i.e., at 
normal temperature, and with good oil supply from cups or pads, 
are as follows : * 

(a) Starting from rest with constant load, the coefficient of 
friction first increases slightly with increasing velocity and then 
decreases, until at a velocity somewhere below 200 feet per min- 
ute (and depending upon the oil supply) a minimum value is 
reached (see Fig. i6).| With further increase of velocity the 

* See Archbutt and Deeley, page 58, and Thurston's " Friction and Lost 
Work," pages 296-312. 

t It is to be noted that this discussion and the coefficients given refer to 
circular bearings and friction of rotation. 



104 



MACHINE DESIGN 



coefficient increases till the temperature affects the viscosity of 
the lubricant to such an extent that abrasion and failure occur. 

(b) With constant velocity and very light loads (see Fig. 17) 
the coefficient of friction is very high. As the load is increased, 
the coefficient decreases very rapidly at first, and then more 
slowly till pressures of about 100 to 200 lbs. per square inch are 
obtained when the coefficient again slowly increases. 

(c) The law of variation of friction with temperature is very 
complex and not well defined. Its general characteristics, how- 
ever, may be expressed as follows: every combination of pres- 
sure and velocity requires a lubricant of a certain viscosity for 
best results. At high speeds and light loads, a light, thin oil 



600 



500 



o 
3400 

a 
-300 



®200 



100 





/ 












/ 












/ 

/ 












/ 



































,005 .01 .015 .02 .025 
Coefficient of Friction 

Fig. 16. 



.03 



000 



500 



400 



2,300 

OS 

g200 



.100 















































. 






\ 























.005 .01 .015 .02 .025 
Coefficient of Friction 

Fig. 17. 



.03 



will be readily drawn in between the bearings, and its fluid fric- 
tion, which constitutes the greater part of the resistance in such 
cases, will be less than that of a heavier oil. Increasing the 
temperature of a lubricant decreases its viscosity and, in the 
above case therefore, would cause a decrease in friction. In 
the case of the heavier loads and lower velocities, usually met with 
in machines, an increase of temperature decreases the viscosity 
and may, owing to the expulsion of the lubricant, give an increase 
in friction. 

Care should therefore be used to obtain an oil suited to the 
case in hand, for sometimes a change of lubricant is suffi- 



GENERAL THEORY OF FRICTION 105 

cient to cause great trouble or, on the other hand, to reduce the 
temperature of a bearing that is heating. The failure of 
imperfectly lubricated bearings generally results from the 
lowering of the viscosity by increased temperature, so that the 
oil film is no longer maintained and metallic contact and abra- 
sion ensue. 

From the foregoing it is evident that the coefficient of friction 
for imperfect lubrication will necessarily be a variable quantity. 
Figs. 16 and 17 show the variation of fi for varying velocities 
and pressures. With good lubrication and moderate velocity it 
may be as low as .005, and again with low velocity and poor 
lubrication it may rise to .05 or more. When the velocity is 
exceedingly low, the coefficient approaches that of static friction 
of lubricated surfaces, the average value of which is .15. A fair 
average range for pressures from 50 to 500 lbs., and velocities 
from 50 to 500 ft. per minute, is from .02 to .008 and, for pur- 
poses of design of ordinary machinery, may be taken at .015. It 
is to be noted that with imperfectly lubricated surfaces and low 
velocities the coefficient of friction is less dependent on the 
character of the lubricant, and more dependent on the character 
of the rubbing surfaces. The curves Figs. 16 and 17 are com- 
posite curves taken from a number of actual experimental re- 
sults. They are not to be taken as giving exact values of the 
coefficient /*, but serve to show graphically the general laws by 
which it varies. In interpreting such curves as Fig. 17 it must 
be kept in mind that, while the coefficient is decreasing or in- 
creasing, the actual frictional resistance may not be changing in 
like manner. The frictional resistance is the product of the 
load and the coefficient of friction. If, for instance, the coeffi- 
cient decreases as fast as the load increases, the frictional resist- 
ance will remain constant. The curves show, however, where 
best results may be expected when designing new machinery, 
and throw some light on proposed changes in running speed of 
machinery already installed. They also indicate the complexity 
of the relation which exists between velocity, pressure, and the 
coefficient of friction. When it is considered that the tempera- 
ture also greatly affects these relations, it is evident that a state- 



106 MACHINE DESIGN 

ment of these relations for imperfect lubrication, in the form of 
a general law or mathematical expression, is impracticable, and 
all such expressions are misleading. 

33. Perfect Lubrication. It has been shown in the last 
article that any rotating journal will, by means of the molecular 
attraction between it and the lubricant, combined with the 
viscosity of the lubricant, draw more or less of the lubricant in 
between the journal and bearing, the amount so drawn in de- 
pending on the velocity and pressure. If the journal be allowed 
to run in an oil bath, or is otherwise plentifully supplied with 
oil, and the velocity be high enough for the pressure carried, it 
is found that this action is so marked that the rubbing surfaces 
are completely separated by a thin film of lubricant and the 
friction becomes only that due to the fluid friction of the lubricant 
itself. 

Mr. Beaucamp Tower experimenting with journal friction (see 
Proceedings of Institution of Mechanical Engineers, 1883) found 
that with a journal and bearing arranged as in Fig. 15, the 
above action was so marked as to form a film of oil under pres- 
sure such that the load was completely fluid borne. The distri- 
bution of the pressure in this film was found to be as indicated 
by the diagrams above the cross-sections, rising to a maximum 
at the middle and falling to zero at the edges of the bearing. 
Mr. Tower succeeded in this way in carrying a load of 625 
pounds per square inch of projected area at a velocity of 471 ft. 
per minute. With a load of about 330 lbs. per sq. inch, and a 
velocity of about 150 ft. per minute, a maximum oil pressure of 
625 lbs. was found near the middle point of the bearing. It has 
been proved mathematically, and verified experimentally, that 
the conditions which exist in a bearing running under these 
conditions are as follows: the journal, being slightly smaller 
than the bore of the bearing, tends to be crowded back from the 
side where the lubricant is carried in, as shown in an exag- 
gerated manner in the figure, giving a wedging effect. The 
pressure is consequently greatest at a point a little more than 
half way beyond the centre of loading where the distance be- 
tween surfaces is least. 



GENERAL THEORY OF FRICTION 



107 



The exact relation which must exist between velocity and 
pressure, to allow this pressure film to form, is not known nor 
is it likely that exact limits can ever be set. Enough is known, 
however, to serve as a general guide for average conditions. 

Professor H. F. Moore found that for circular journals the 
minimum limiting values of pressure and velocity where the film 
will just form may be approximately expressed by the expression 
w— 7.47 \As* where wis in pounds per square inch, and v in feet 
per minute. The values given by this expression are plotted in 
Fig. 18, Curve No. 1, and seem to check fairly well with con- 
siderable other data. Curve (2), Fig. iS, shows the simultaneous 
values obtained by Tower with olive oil, where frictional resist- 
ance was a minimum, indicating that the film was at least well 



800 



£200 
u 

<D 

G. 

1 100 
o 





















No. 2 






No. 3 




















p*i 


.W^L- 




No. 1- 



























BO 100 150 200 250 300 

Feet per "Minute 

Fig. 18. 



350 



,400 



450 500 



formed. Curve (3) shows similar values for mineral oil. The 
values obtained by Moore are on the safe side judged by Tower's 
work, which is accepted as accurate, and probably do not indicate 
the very lowest point at which a film will form. Tower found 
that a film would form considerably below the values given in 
Curve (2). In Moore's experiments, as in Tower's, the tempera- 
ture was constant at 90 . Moore's experiments were on mineral 
oils. The results of Tower's experiments are very concordant 
and conclusive, and show that the laws of friction for perfectly 
lubricated surfaces, for ordinary speeds and pressures, are quite 
definite, the coefficient of friction varying as the square root 
of the velocity and inversely as the pressure, very nearly. 



* See American Machinist, Sept. 16, 1903. 



io8 



MACHINE DESIGN 



Thus for olive oil the relation is expressed very closely by 

y/ ' v 
li = .2 . It follows from this, that for any fixed velocity and 

w 

temperature the product of ft and w will be a constant. That is, 
the frictional resistance is practically constant with change of 
load, for any velocity. This was actually found to be the case in 
the experiments, a variation of pressure per square inch from ioo 
to 500 not appreciably affecting the resistance. Table VIII will 
serve to show the remarkable regularity of the results, and the low 
values of the coefficient as compared with imperfectly lubricated 
surfaces. Much lower values have since been attained in oil- 
testing machines, under more ideal conditions, but such low 
values must not be considered as attainable under ordinary prac- 
tical working conditions, while there is no good reason why such 
coefficients as given below cannot be obtained in well-constructed 
machinery. 

TABLE VIII 

BATH OF RAPESEED OIL 



Load in 

lbs. 
per sq. 
Inch. 


Coefficients of Friction for Speeds as Below. 


105 ft. 
per Min. 


157 ft. 
per Min. 


209 ft. 
per Min. 


262 ft. 
per Min. 


314 ft. 

per Min. 


366 ft. 
per Min. 


419 ft. 
per Min. 


47i ft. 
per Min. 


573 
520 

4i5 
363 
258 

i53 

IOO 


.00107 
.00162 
.00277 


.00102 
.00095 
.00093 
.00084 
.00139 
.00200 
•°°357 


.00108 
.00105 
.00107 
.00960 
.00162 
.00239 
.00423 


.00118 
.00115 
.00119 
.00110 
.00178 
.00267 
.00503 


.00126 
.00125 
.00130 
.00122 
.00195 
.00300 
.00576 


.00132 
.00133 
.00140 
.00134 
.00213 
•00334 
.00619 


.00139 
.00142 
.00149 
.00147 
.00227 
.00367 
.00663 


.00148 
.00158 
•00155 
.00243 
.00396 
.00714 



Tower's experiments have been amply verified and may be 
accepted as reliable for the range which they cover. The ex- 
periments of Stribeck and Lasche (see Chap. X) have extended 
the range of knowledge on this point to velocities over 2,000 ft. 
per minute. Their experiments show that for velocities between 
500 and 2,000 ft. per minute the coefficient of friction, for a given 
load and temperature, varies as the 5th root of the velocity; and 
beyond 2,000 ft. is independent of the velocity. This point is 



GENERAL THEORY OF FRICTION log 

discussed still further in Chap. X in connection with the design 
of bearings, where its principal application is found. 

34. Summary. From the foregoing discussion the following 
statements may be made: 

(a) The friction of imperfectly lubricated surfaces depends 
partly on the character of the surfaces themselves, and in a greater 
degree on the character and amount of the lubricant supplied. 

(b) The load that can be successfully carried on an imperfectly 
lubricated surface will vary greatly with the amount of lubricant 
supplied, and must be kept very low where this supply is re- 
stricted. 

(c) The friction of perfectly lubricated surfaces depends very 
little on the character of the rubbing surfaces, but depends 
mainly on the character of the lubricant. 

(d) The frictional resistance of perfectly lubricated surfaces 
is, within the ordinary limits, independent of the intensity of 
pressure and dependent only on the velocity. 

(e) The coefficient of friction of perfectly lubricated surfaces, 
for any given pressure and temperature, varies very nearly as the 
square root of the velocity for velocities up to 500 ft. per minute; 
approximately as the fifth root of the velocity for velocities be- 
tween 500 and 2,000 ft. per minute; and is practically independent 
of the velocity for values above 2,000 ft. per minute. 

35- Efficiency. It has been pointed out that all the energy 
supplied to a machine is not transformed into useful work, but 
that some of it is always lost in overcoming frictional resistances 
and doing useless work. There are many ways in which energy 
losses may occur in machines, and a careful distinction must be 
made between certain of these ways in order to get a clear 
definition of the term efficiency. Thus the steam engine re- 
ceives its supply of heat in the form of steam under pressure. 
A considerable portion of the heat so received is lost by con- 
densation of steam on the cooler cylinder walls, and some escapes 
by radiation without doing any work whatever on the piston. 
Of the energy actually applied to the piston, part is transformed 
into useful work at the driving belt, and part is lost in over- 



IIO MACHINE DESIGN 

coming the frictional resistances just discussed at the various 
constraining surfaces. 

The gas engine is subject to similar losses; a large part of the 
heat of combustion escaping to the jacket water or to the at- 
mosphere by radiation, and doing no work on the piston; while 
only a part of the energy actually applied to the piston reappears 
as useful work. Hydraulic and electric machinery have similar 
elements of loss. The first class of these energy losses might be 
called leakage losses, as they are of the same character as losses 
by actual leakage of the medium which is used to transmit the 
energy. The losses in the machine itself are known as frictional 
losses and are common to all machines; and no machine can 
transform all the energy supplied into useful work, but must lose 
some of it in friction or other wasteful resistances. 

Efficiency has been defined (Art. 2) as the ratio of useful 
work to energy supplied; and from the above it appears that a 
machine may have two efficiencies depending on whether refer- 
ence is had to total energy supplied, or to that portion only of 
the total energy which the machine transforms into useful and 
useless work. These efficiencies are respectively known as the 
Absolute Efficiency and the Mechanical Efficiency. Thus, 
if a gas engine is supplied with 1,000 thermal units, and trans- 
forms 200 units into useful work, and 50 units into the useless 

work of friction, its absolute efficiency is -^- = .20, and the 

1,000 
200 

mechanical efficiency is — = .80. The consideration of abso- 

250 

lute efficiency is beyond the scope of this work; for the design 

of many machines it does not need to be considered; but the 

mechanical efficiency can seldom be neglected, since, in general, 

the amount of work to be done is fixed, and the source of energy 

must supply enough more energy than this to compensate for the 

frictional losses of the machine. 

The mechanical efficiency of any train of mechanism is the 

continued product of the efficiencies* of all the several pairs of 

* It may be noted in passing that the term efficiency is used in a number of ways 
other than as the ratio of work done to energy expended. Thus the strength of a 



GENERAL THEORY OF FRICTION III 

constraining surfaces in the train at which frictional losses 
occur. Let any machine have n pairs of such surfaces, and let 

their respective efficiencies be e, e v e 2 , e 3y e iy e n . Let E 

be the mechanical efficiency of the whole machine, and let K be 
the total amount of energy available for transformation into 
either useful or useless work. Then, the amount of energy which 
the first pair of constraining surfaces delivers to the second is 
K X e, and the amount which the second delivers to the third is 
Ke X e lt and so on, till the amount of energy delivered by the 

last element (or the work done) is K (e X e t X e 2 ej. 

But the mechanical efficiency of the train is 

work done K (e X e x X e 2 e n ) 

energy supplied K 

= (e X e, X e 2 ej. 

A machine may consist of several trains of mechanism. If 
these several trains are arranged in series so that the energy 
passes from one to another consecutively, the efficiency of the 
whole machine, by reasoning similar to that in the last paragraph, 
is the continued product of the efficiencies of the several trains 
of mechanism. If, however, the trains are arranged in parallel 
so that the total energy is transmitted simultaneously through 
several trains of mechanism, each train transmitting only a por- 
tion of the energy, the above reasoning for the efficiency of the 
whole machine does not hold. If the amount of energy supplied 
to each train is known, the amount of work which it will deliver 
can be computed as above. The sum of all the work, delivered 
by all the trains, divided by the total energy supplied, will be the 
efficiency of the whole machine. 

If, therefore, the efficiencies of the several constraining sur- 
faces of a machine are known, the mechanical efficiency of the 

riveted joint, compared to the strength of the original unpunched plate, is called 
the efficiency of the joint, when what really is meant is its relative strength. Again, 
in an air compressor, the ratio of the air actually discharged per stroke, to the whole 
amount raised to the required pressure per stroke, is called the volumetric efficiency. 
It is evident that such efficiencies are of a different character from those discussed 
above and do not enter into the calculations of the efficiency of the machine, as a 
whole, in the manner indicated above. 



112 MACHINE DESIGN 

whole machine can be calculated. The mechanical efficiency of 
any machine element is, however, a variable quantity; for the 
coefficient of friction of any pair of constraining surfaces will 
vary with the lubricant and its method of application, the tem- 
perature, the alignment of the surfaces, the velocity of rubbing, 
and the bearing pressure. Furthermore, when all other condi- 
tions are constant, the same pair of constraining surfaces will 
have an entirely different efficiency for the same amount of 
power transmitted, depending on the manner in which the load 
is applied. Thus, consider a simple wheel and axle driven by a 
belt on the periphery of the wheel. With a given diameter of 
wheel, the transmission of a given amount of power will bring a 
certain definite frictional load on the bearings. If, however, 
the diameter of the wheel is doubled, the belt speed is increased 
in a like ratio, and the belt tension will, for the same power 
transmitted, be one-half of the former value; and, as a conse- 
quence, the frictional resistance at the bearings will be reduced 
to one-half the original value, the revolutions remaining con- 
stant. 

In general, therefore, it is impossible to calculate precisely 
from the analysis of a design what the mechanical efficiency 
will be, particularly if the mechanism is at all complicated, 
though a reasonable approximation is possible. If machines of 
a similar type have been built, it is far more accurate to base 
the design of new ones on efficiency tests made on those already 
in existence. For all standard machines such tests have been 
made, and the recorded results form a valuable basis for the de- 
sign of new machines of like characteristics. But when a ma- 
chine of a new type is to be designed, and no recorded tests are 
to be had that will give any information as to the probable ef- 
ficiency, an estimate must often be made and the efficiency cal- 
culated as outlined above. In general, a close approximation 
can be made, and the making of such estimates is a great aid to 
the development of that judgment in such matters, which comes 
only with experience. In such cases a knowledge of the ef- 
ficiencies of various machine elements becomes necessary. If 
the coefficient of friction for any constraining surface could be 



GENERAL THEORY OF FRICTION 113 

accurately determined, it would be possible to calculate its 
efficiency with some degree of certainty. But, as before noted, 
the quantity varies with the velocity of rubbing, with changes in 
bearing pressures, etc., and such methods of computation are 
necessarily cumbersome and to be attempted only where a very 
close estimate is required. 

The following are rough average values of the efficiencies of 
the most common elements. For more accurate values the stu- 
dent is referred to the respective discussions of these various 
elements which follow : 

Common Bearing, singly 96-98 

Common Bearing, long lines of shafting 95 

Roller Bearing 98 

Ball Bearings 99 

Spur Gear Cast Teeth, including bearings 93 

Spur Gear Cut Teeth, including bearings 96 

Bevel Gear Cast Teeth, including bearings 92 

Bevel Gear Cut Teeth, including bearings , 95 

Worm Gear, varies with thread angle, see Art. 54 

Belting 96-98 

Pin-connected Chains, as used on bicycles 95~97 

High Grade Transmission Chains 97~99 

S 



CHAPTER V 
SPRINGS 

36. Distinguishing Characteristic of Springs. Springs are 
characterized by a considerable distortion under a moderate load. 
Every machine member is, in a sense, a spring, for no material 
is absolutely rigid and the application of a load always produces 
stress and accompanying strain. By proper selection and distri- 
bution of material it is possible to control (within wide limits) 
the degree of distortion under a given load. 

An absolutely rigid material would be practically unfit for the 
construction of any member subject to other than a perfectly 
quiescent load; for (as shown in Art. 24) the stress due to a sud- 
denly applied load would be infinite if the corresponding distor- 
tion of the member were zero. 

While it is usually desirable to restrict the distortions 
of most machine parts to very small magnitudes, there are 
many cases in which considerable distortion under moderate 
load is desirable or essential. To meet this last requirement 
the member is often given some one of the forms commonly 
called springs. 

37. The Principal Applications of Springs. Springs are in 
common use: 

I. For weighing forces; as in spring balances, dynamome- 
ters, etc. 

II. For controlling the motions of members of a mechanism 
which would otherwise be incompletely constrained ; for example, 
in maintaining contact between a cam and its follower. This 
constitutes what Reuleaux has called " force closure." 

III. For absorbing energy due to the sudden application of a 
force (shock) ; as in the springs of railway cars, etc. 

IV. As a means of storing energy, or as a secondary source of 
energy; as in clocks, etc. 

114 



SPRINGS 



"5 



An important class of mechanisms in which springs are used 
to weigh forces is a common type of governor for regulating the 
speed of engines or other motors. In those governors which use 
springs to oppose the centrifugal, or other inertia actions, the 
springs automatically weigh forces which are functions of speed, 
or of change of speed. The links, or other connections, which 
move relative to the shaft with any variation of the above forces, 
correspond to the indicating mechanism of ordinary weighing 
devices. 

The first of the above-mentioned applications — the weighing of 
forces — is usually the most exacting as to the relation between 
the load and the distortion of the spring throughout the range of 
action. In the second and third classes of application, it is fre- 
quently only required that the maximum load and distortion 
shall lie within certain limits, which often need not be very pre- 
cisely defined. The use of springs for storing energy (as the 
term spring is ordinarily understood) is almost wholly confined to 
light mechanisms or pieces of apparatus requiring but little 
power to operate them. 

38. Materials of Springs. Springs are usually of metal; 
although other solid substances, as wood, are sometimes used. 
A high grade of steel, designated as spring steel, is the most 
common material for heavy springs, but brass (or some other 
alloy) is often used for lighter ones. 

A confined quantity of air, or other compressible fluid, is used 
in many important applications to perform the office of a spring. 
The air-chamber of a pump with its inclosed air is a familiar ex- 
ample of what may be called a fluid spring used to reduce shock 
("water hammer"). The characteristic distortion of the solid 
springs is a change in form rather than of volume; while the 
fluid springs are characterized by a change of volume with inci- 
dental change of form. 

Soft-rubber cushions, or buffers, are not infrequently employed 
as springs, and these are in some respects intermediate in their 
action to the two classes mentioned above. It is usually not nec- 
essary, in these simple buffers, or cushions, to secure a very exact 
relation between the loads and the distortions under such loads. 



n6 



MACHINE DESIGN 



The discussion of the confined gases (fluid springs) is not within 
the scope of the present work; hence the following treatment 
will be limited to solid springs. 

39. Forms of Solid Springs. Springs may be subjected to 
actions which extend, shorten, twist, or bend them, producing 




Fig. 19. 



^ 



Fig. 22. 



1000 



£7 



Fig. 25. 



— — 


















I 
1 








el- 













KSS2^> 



^*r 




stresses in the material, the character of which depends both upon 
the form of the spring and upon the manner of applying the load. 
I. Flat Springs are essentially beams, either cantilevers, or 
with more than one support. These springs are subjected to 
flexure when the load is applied, and the resultant stresses are 
tension in certain portions of the material, and compression in 
others, with a transverse shear, as in all beams; the shear may 



SPRINGS 



117 



usually be neglected in computations. The ordinary beam 
formulae for strength and rigidity may be applied to flat springs, 
with constants appropriate to the particular material and form of 
beam used. 

Flat springs may be simple prismatic strips, of uniform cross- 
section (Fig. 19 or 22), although it is preferable that the form 
of such springs approximate those of the "uniform strength" 
beams (Figs. 20 or 21; 23 or 24). 

It is often desirable or practically necessary to build up these 
springs of several layers, leaves, or plates, producing a laminated 
spring. It will appear from the discussion of these laminated 





Fig. 30. 



Fig. 30 (a). 



springs that they may be properly treated as a modification of one 
form of " uniform strength" beam. The neutral surface of the 
beam used as a spring may be initially curved, either to clear 
other bodies, or to give the spring an advantageous form when it 
is under normal load. See Fig. 27. 

Two or more springs may be compounded, as in the "ellipti- 
cal" springs or in the platform springs frequently used under 
carriages. In such cases, each spring may be computed sepa- 
rately, and the total deflection is the sum of the deflections of the 
separate springs of the set. 

II. Helical, or Coil Springs are most commonly used to resist 
actions which extend, shorten, or twist the spring relatively to 
its longitudinal axis. These are sometimes improperly called 
spiral springs. 



Il8 MACHINE DESIGN 

The stress in the wire (or rod) of which a helical spring is 
made is somewhat complex, consisting of torsion combined with 
tension or compression, or both. In a "pull spring," one which 
is extended longitudinally under the load, the predominating 
stress (with ordinary proportions) is a torsion, and there is a 
secondary tensile stress in the wire. In a "push spring,' ' one 
which is shortened by the load, the predominating stress is tor- 
sion, with a secondary compressive stress. When the helical 
spring is subjected to an action which twists the spring (as a 
whole) the principal stress in the wire is that due to flexure 
(tension and compression in opposite fibres) and the secondary 
stress is torsion. 

Helical springs are sometimes arranged in "nests," springs of 
smaller diameter being placed within those of larger diameter, 
(Fig. 30) . In these cases, the different springs of a set are com- 
puted separately. This last arrangement is common practice in 
car trucks. 

III. Spiral Springs, properly so called are those of the form of 
the familiar clock spring. These are best adapted for a twist 
relative to the axis of the spiral, and are usually employed when 
a very large angle of torsion between the two connections is 
necessary. In this form of spring, the stress in the material is 
that due to flexure : or tensile and compressive stress on opposite 
sides of the neutral axis. 

IV. Helico-Spiral Springs. The form of spring represented 
by the common upholstery spring may be looked upon as a spiral 
spring which has been elongated, and given a permanent set, in 
the direction of its axis; or it may be considered as a modified 
helical spring in which the radii of the successive coils are not 
equal. It is thus intermediate between the two preceding classes. 
This last form is not usual in machine construction; though it 
has the advantage over the common helical spring of con- 
siderable lateral resistance, and it may be employed to 
advantage where it is difficult or undesirable otherwise to 
constrain the spring against buckling. This spring is used only 
as a push spring, to resist a compressive action. The springs 
used on the ordinary disc valves of pumps are often of this 



SPRINGS 



II 9 



form, as they will close up flat between the valve and guard. 
Car springs are sometimes made of a flat strip or ribbon of steel 
wound in this general form, with the edges of the strip parallel 
to the axis of the spring. 

V. Occasionally straight rods, usually of circular or rectangular 
cross-sections, are employed to resist torsion relative to their 
longitudinal axis. These are comparatively stiff springs, and 
the stress is, of course, torsional. Every line of shafting is 
necessarily a spring, in this sense. 

The following summary gives the ordinary forms of solid 
springs; the kinds of loading to which they are subjected; and 
the predominating stresses resulting from the different loads. 





GENERAL SUMMARY OF 


SPRINGS 


Form of Spring. 


Load Action. 


Predominating Stress. 


Flat Spring. 
Helical Spring. 

(« <c 
<( (« 

Spiral 


Flexure or Bending. 
Extension, Pull. 
Compression, Push. 
Torsion, Twist. 
Torsion, Twist. 


Tension and Compression. 
Torsion (plus). 
Torsion (minus). 
Tension and Compression. 
Tension and Compression. 



41. Computations of Simple Flat Springs. The following 
notation will be used in treating of flat springs with rectangular 
cross-sections. 

P = load applied to the spring. 

/ = free length of the spring. 
p = intensity of stress in outer fibres. 

/ = moment of inertia of most strained section. 

h = dimension of this section in plane of flexure. 

b = dimension of this section perpendicular to plane of flexure. 
E = modulus of elasticity of material. 
d = deflection of the spring. 

The six forms of rectangular section beams, shown by Figs. 19 
to 24, are the most important of those used as simple flat springs. 
These will be designated Type I, II, etc., as in the following 



120 



MACHINE DESIGN 



table, which gives the constants to be substituted in the general 
formulae for computations relating to each type. 



TABLE IX 





Type. 


Coefficients. 


A 


fi 


B 


K 


c 


I 


As per Fig. 19 


2 
3 


1 

4~8 


1 
4 


1 

6 


3 
2 


II 


" " " 20 


2 
3 


1 


3 

8 


1 
4 


3 


III 


" " " 21 


2 
3 


1 

"2T 


1 


1 


3 
2 


IV 


" " " 22 


1 
6 


1 
3 


4 


2 
3 


6 


V 


« « « 23 


1 
6 


1 


6 


I 


6 


VI 


« « « 24 


1 

6 


2 
3 


8 


4 


6 



The theory of strength against flexure (equation J and tables 
1 and 2) gives: For rectangular section beams supported at the 
ends and loaded at the middle (Types I, II, III). 

-Pl = ?-pbh 2 .\Pl = -pbh 2 . . . (1) 

For the rectangular section cantilevers, with load at free end, 

Pl = ^pbh 2 (2) 

Or the general formula for the strength of rectangular section 
beams may be written 

PI = Apbh 2 (3) 

In which the coefficient A has the values given in the Table. 
The theory of elasticity of beams gives 

PI 3 
8 ~P--ET> M 

or for rectangular cross-sections 

PI 3 

'-'m* ■ (5) 

In which p and B are as given in the Table, for the types under 
consideration. 

The last equation (5) may be used for all computations as to 
rigidity of flat springs (beams), provided the elastic limit is not 



SPRINGS 121 

exceeded. The only constant for the material which enters this 
expression is the modulus of elasticity (E); this is simply the 
ratio of stress to strain which holds up to, but not beyond, the 
elastic limit; hence any computation made by this formula 
should be checked for safety. Equation (3) may be used for this 
purpose. To illustrate, assume that a rectangular section pris- 
matic spring (Type I) has a length between supports of 1 = 30" '; 
the load at the middle is P = 1,000 lbs.; the deflection under this 
load is to be $ = 1 . 5 inches ; and the spring is made of a single 
strip of steel Y% inch thick (h) . Required the breadth (b) of the 
spring, assuming the modulus of elasticity, £ = 30,000,000. 
From eq. (5) : — 

_ PI 3 1 1,000 X 27,000 X 512 rt . , 

b = B -=—j- 3 = - X — = 2.84 + inches. 

Edh 6 4 30,000,000 X 1.5 X 27 

This gives the width of spring for the required relation of the 
deflection to load; that is, it gives a spring of the required stiff- 
ness, provided the stress produced does not exceed the elastic 
limit. It is necessary to check the spring as found above, for if 
the elastic stress is passed, the spring not only takes a permanent 
set, but the required ratio of the load to the deflection will not be 
secured. On the other hand, it is often important for economy 
of material to use as light a spring as is consistent with safety; 
or, in other words, it is important not to have too low a working 
stress under the maximum load. 

From eq. (3) : — 

PI 3 X 1000 X 30 X 64 

p = -rrn — n = 112,500 lbs. per sq. inch. 

r Abhr 2 x 2.84 X 9 

This stress is beyond the elastic limit of any ordinary grade of 
steel, hence it is probable that some different form of spring should 
be used. A change could be assumed, as in the thickness of the 
plate, and new computations made with the new data. A thinner 
plate would reduce the stress, but it would demand a wider spring 
for the required stiffness. A more general method will now be 
given, by which it is possible to determine the proper spring for 
given requirements without the necessity of successive trial com- 
putations. 



122 MACHINE DESIGN 

From eq. (3) : — 



PI Plh 

bh 2 = -—.'. bh 3 = —-— .... (6) 
Ap A p w 



From eq. (5) :— 

BPP 



" 3 = ~ET (7) 



From eqs. (6) and (7) : — 

Plh BPl 



A p E d 

pp pP 

k - AB h- K h • • • • (8) 



From eq. (3) :• 



, 1 PI r Pl , . 

h = TJv = c Jv ' ' ■ ' (9) 

The two equations (8) and (9) are in convenient form for de- 
signing a flat spring when the span (/), deflection (#), load (P), 
and the material are given. Example : The span of a rectangular 
section prismatic flat spring (Type I) is 30 inches; and a load of 
1,000 lbs. applied at the middle is to cause a deflection of 1.5 
inches. 

If the modulus of elasticity be 30,000,000 and the safe maxi- 
mum working stress be taken at 50,000 lbs. per sq. in.,* required 
the dimensions of the cross-section, h and b. 

From eq. (8) : — 

-rr PP 1 50,000 X QOO I . , 

h = K-~ = - X — — = ~ inch. 

ho 6 30,000,000 X 1.5 6 

Taking h = ^ mcn > to use a regular size of stock, p will be 
somewhat less than 50,000, or 

<; I 
p : 50,000 :: -^2 : jrj . ' . p = 47P 00 - 

From eq. (9) : — 

6 = C— -'3- x IoooX 3°Xio24 - , - inches 
p h 2 2 47,000 x 25 ° y 

* If the spring is provided with stops to prevent deflection beyond a certain 
amount, the stress due to such deflection may be nearly equal to the elastic limit of 
the material. A very small factor of safety is all that is necessary. 



SPRINGS 123 

If this width is inadmissible, a laminated or plate spring may 
be used. See next article. 

It will be noted that equation (8) does not directly involve 
either the load P or the breadth of spring b. It is evident that if 
a beam (flat spring) of given span (/), and thickness (h), is 
caused to deflect a given amount (#) , the outer fibres will undergo 
a definite strain which is not dependent upon the width of the 
beam (b), nor upon the force required to produce this change in 
relative positions of the molecules. As the unit strain multiplied 
by the modulus of elasticity equals the unit stress, it follows that 
this stress may be computed from /, h, and d (which determine 
the strain), in connection with E. If the breadth of the beam 
(b) is increased, the force (P) required to produce the given de- 
flection (£) will be proportionately increased, but the intensity of 
stress is not affected by these changes alone. 

This same conclusion may be reached from the following rela- 
tion,* in which p = the radius of curvature due to load. 

EI pi Eh 

P = EI^ 1 —= . . . (10) 

M y 2 h 2 p v } 

Eh , . 

2 p 

It appears from eq. (11) that the stress is simply proportional 
to the thickness (h) and the radius of curvature (p), for any 
given value of E. The span /, and the deflection d y determine p, 
so that eq. (10) or (11) may take the place of eq. (8). Equations 
(10) and (11) are important in connection with the theory of 
laminated springs. 

42. Laminated, or Plate, Springs. It was shown in the preced- 
ing article that the maximum thickness of a simple flat spring is 
fixed vvhen the span, deflection, and modulus of elasticity are 
known, and the intensity or working stress has been assigned. 
[See eq.(8).] With the value of the thickness (h) thus limited 
it will frequently happen that a simple spring will require ex- 
cessive breadth (b) to sustain the given load, and it is often 
necessary to use a spring built up of several plates or leaves. 

* See Church's " Mechanics," page 250. 



124 MACHINE DESIGN 

Example: P = i,ooo lbs.; / = 3o"; p = 60,000 lbs. per sq. 
in. ; <5 = 2", and E = 30,000,000. A simple prismatic spring of 
rectangular section, with load at the middle of the span (Type I), 
to meet the above requirements would have: 

Tr PP 1 60,000 X QOO . , 

h = K -§— = - X — — = .15 inch. 

ho 6 30,000,000 x 2 

b = C —- 2 = — X 2 — " = ^VA inches. 

p h 2 2 60,000 x .0225 °° 

This spring, consisting of a plate .15 inch thick and 33^ 
inches wide, with a span of 30 inches, is evidently an impractica- 
ble one for any ordinary case. Suppose this plate be split into six 
strips of equal width, each 33.3 -r-6 = 5-5' r wide, and that these 
strips are piled upon each other as in Fig. 25; then, except for 
friction between the various strips, the spring would be exactly 
equivalent* as to stiffness and intensity of stress, to the simple 
spring computed above. While the form of laminated spring 
which has just been developed might answer in some cases, an- 
other form, based upon the " uniform strength" beam (Type II), 
is much better for the ordinary conditions. It may be developed 
as follows, taking the same data as the preceding example except 
that the spring is to be of Type II, Fig. 20. 

In the simple spring, Type II, Table IX 

Tr pP I 60,000 X QOO . , 

h = K-=— = — X = .22 s inches. 

ho 4 30,000,000 x 2 

1 n P l 3 ^ 1,000 X 30 . 

b = C . — 7^ = - X ; 7 = 14.8 inches. 

p h 2 60,000 X .0506 

A laminated spring for the case under consideration may be 
derived from this simple spring by imagining the lozenge-shaped 
plate to be cut into strips which are piled one upon another as in- 
dicated in Fig. 26. The thickness of .225 inches does not corre- 
spond to a regular commercial size of stock, however, and it will 
usually be better to modify the spring to permit using standard 
stock. If a thickness of %" be assumed for the leaves or plates, 
the stress, as found from eq. (8) of the preceding article becomes: 



SPRINGS 125 

hEd 4 X .25 X 30,000,000 X 2 

P = -^TTT = : = 66,700. 

r i£r 900 

If this stress is considered too great, steel jq" thick might be 

4 X 3 X 30,000,000 X 2 

used, when p = - = ^0,000. 

r 16 X 900 J 

With h = T V, and p = 50,000, 

_, P/ -i 1,000 X 30 X 2=56 

6 = C —^ = - X = 25.6". 

p hr 2 50,000 x 9 

If this spring, 30" span, -^" thick, and 25.6" wide at the 
middle, be replaced by 5 equivalent strips, each 25.6 -r- 5 = 5.11" 
wide (nearly $}i"), see Fig. 26, a laminated spring of good form 
and practical dimensions will result. In cases where the maxi- 
mum allowable width of spring is fixed, a larger number of 
plates may be necessary. Thus, in the preceding problem, if the 
spring width must be kept within 4%", it is necessary to use 6 
plates, each 25.6 + 6 = 4. 2 7" wide. In actual springs, the usual 
construction is that shown by Fig. 27, in which the several plates 
have the ends cut square across instead of terminating in tri- 
angles. These springs approximate uniform strength beams, and 
may be computed by equations (8) and (9) of Art. 41, remember- 
ing that b is the breadth of the equivalent simple spring. Or, if 
n is the number of plates and b t the breadth of each plate in the 
laminated spring, nb l = b. 

The last of these formulae, eq. (9), is not strictly applicable 
when the ends of the plates are cut square across; but it may 
generally be used with sufficient accuracy, provided the succes- 
sive plates are regularly shortened by uniform amounts. It is 
quite common practice to have two or more of the plates extend 
the full length of the spring. This construction makes the spring 
a combination of the triangular and prismatic types (Type II 
and Type I, or Type V and Type IV, depending upon whether 
the spring is supported at the ends, or is a cantilever). Mr. G. 
R. Henderson in discussing the cantilever form (Trans. A. S. 



126 MACHINE DESIGN 

M. E., Vol. XVI), says:— -"For a spring with all the plates full 
length we would have (see eq. 5) 

En\h 3 

so for one-fourth of the leaves full length, the deflection would be 
decreased approximately one-fourth of the difference between 

6P1 3 , 4PI 3 S.5P/ 3 " 

and „ , , or 



Enb x h 3 Enbh 3 Enbh 3 

By similar reasoning, for a spring loaded at the middle and 
supported at the ends, with one-fourth the plates extending the 
whole length of the spring, 

32 E n b x h 3 ' 

This may be otherwise stated as follows: 

When the number of full-length leaves is one-fourth the total 
number of leaves in the spring, use ±± B instead of B and ii K 
instead of K in equations (5) and (8) of the preceding article; 
the values of B and K being those given for the triangular forms, 
Type II or Type V, as the case may be. 

The spring shown in Fig. 27 is initially curved (when free), 
which is common practice. The best results are obtained by 
having the plates straight when the spring is under its normal full 
load (if this is practicable) because the sliding of the plates upon 
each other, with the vibrations, is then reduced to a minimum. 
The several plates of a laminated spring are usually secured by 
a band shrunk around them at the middle of the span. This band 
stiffens the spring at the middle, and one-half the length of the 
band {% /, Fig. 27) may be deducted from the full span to give 
the effective span to be used as / in the above formulae. It is 
not uncommon to make the longest plate thicker than the others, 
if but one plate is given the full length of the spring. This cannot 
be looked upon as desirable practice, however, as all of the plates 
are subjected to the same change in radius of curvature; hence the 
thicker plate is subjected to the greater stress. See equation (11). 

The following formulae (derived from the preceding) may be 
used in computing flat springs; but it must be remembered that 



SPRINGS 127 

there is always liability of considerable variation in the modulus 
of elasticity, hence such computations can only be expected to 
give approximations to the deflections which will be observed by 
tests of actual springs. These computations will be sufficiently 
exact for many purposes; but when it is important accurately to 
determine the scale of the spring (ratio of deflection to load), 
actual tests must be made. In using these formulae the follow- 
ing rules should be observed. 

I. When the several plates are secured by a band shrunk, or 
forced, over them, one-half the length of the band is to be sub- 
tracted from the length of the spring to get the effective length 
of the spring. 

II. When the plates have different thicknesses, the stress 
should be computed for the plate having the maximum thickness. 

III. If more than one plate has the full length of the spring, 
an appropriate modification of the values of the coefficients B and 
K should be made. Thus, when one-fourth of the total number 
of plates are full length, \\ B and \\ K should be used instead of 
B and K (Type II or V) in equations I, II, III, and IV, below. 

EQUATIONS. 

PP 
9 = B „ ,, 3 (I) 

P = ^P (id 

pP pP 

h = AB k- K k • • (III) 

t = ~KY ( R ) 

Eh 

> = 77 w 

p = Apnb 1 l (VI) 

If 

*=iB • • (VII) 

*-iSf-^ (VIII) 



128 MACHINE DESIGN 

Experience shows that thin plates have a higher elastic limit 
than thick plates of similar grade of material. In the practice of 
a prominent eastern railway company, the values allowed for the 
maximum intensity of stress in flat steel springs are, for : 



Plates 


1 
4 


inch 


thick 


p = 90,000 lbs. sq. in. 


a 


5 
16 


ti 


a 


/> = 84,000 " 


(i 


3 

8 


(I 


a 


p = 80,000 " 


it 


7 
16 


tc 


u 


p = 77,000 " " 


<( 


1 
2 


(( 


n 


^ = 75,000 " 



The above values are satisfied by the equation p = 60,000 + 

7,^00 

— - — , in which h is the thickness of plate in inches. 

These values are for the greatest stress to which the material 
can be subjected, as when the spring is deflected down against 
the stops. 

The modulus of elasticity, E, may vary considerably; but its 
value may be assumed at about 30,000,000 in the absence of more 
definite data. 

In designing a new spring, the value of h\is to be found from 
equation (III); then b x is found by equation (VIII). The other 
formulae are useful in checking springs already constructed, for 
deflection due to a given load, or the reverse; for safety, etc. 

43. Helical Springs. If a rod or wire be wound into a flat 
ring with the ends bent in to the centre, Fig. 28, and two equal 
and opposite forces, + P and — P, be applied to these ends 
(perpendicular to the plane of the ring) as indicated, the rod 
will be subjected to torsion. 

If a longer rod be wound into a helix, with the two ends turned 
in radially to the axis, the typical helical spring is produced. If 
two equal and opposite forces, + P and — P, act on these ends, 
along the axis of the helix, they induce a similar stress (torsion) 
in the rod, but as the coils do not lie in planes perpendicular to 
the line of the forces, there is a component of direct stress along 
the rod. This direct stress increases as the pitch of the coils in- 
creases relative to their diameter; but with ordinary proportions 



SPRINGS 1 29 

of springs, the torsion alone need be considered, when the ex- 
ternal forces lie along the axis of the helix. 

The following notation will be used in treating of helical 
springs of circular wire, subjected to an axial load: 

P = the force acting along the axis. 

r = the radius of the coils, to center of wire. 

d = the diameter of wire. 

p = the maximum intensity of stress in wire (torsion) . 

7 p = the polar moment of inertia of wire. 

£ 8 = the transverse modulus of elasticity. 

<5 = the "deflection" (elongation or shortening) of spring. 

w = the number of coils in the spring. 

/ = the length of wire in the helix = 2 - r n (approximately) . 

Suppose a helical spring under an axial load to be cut across 
the wire at any section, and the portion on one side of this section 
to be considered as a free body, Fig. 29. Neglecting the direct 
stress, equilibrium demands that the moment (Pr) of the external 
force shall equal the stress couple, or moment of resistance 

/ A i(? P . , . \ 

( p — — for circular section ) . 
V 16 ' 

If this free portion of the helix is straightened out, as indicated 
by the broken lines in Fig. 29, till its direction is perpendicular 
to the radial end, it will appear that the moment Pr still equals 

the moment of resistance, — p (P. Since the stress and strain are 

the same in this helix and the straight rod, it appears that the 
energy expended against the resilience is the same in both cases 
(the length of wire affected remaining constant) . Or, as the force 
(P) and the arm (r) are the same in both conditions, the distances 
through which this force acts to produce a given torsional stress 
(p) are equal. If a straight rod of length I is subjected to a 
torsional moment Pr, the angle of twist being « (in * measure) , then 

Pr = 



I 

[See Church's "Mechanics," page 236]. 



I30 ■ MACHINE DESIGN 

The energy expended on the rod is the mean force applied mul- 
tiplied by the distance through which this force acts. If the load 
is gradually applied, this energy is ]/ 2 P r «. In the case of the 
corresponding helical spring, the mean force 04 P) acts through a 
distance equal to the " deflection" of the spring (5), or the energy 
expended is y 2 Pd. As pointed out above, the energy expended 
in the two cases is the same, or 





d 
y 2 Pra = % Pd . \ a = - 

r 


.-. Pr = 


—f- = - x — x — — 

/ r 32 2 x r n 



64 r 2 n 

•P = ^ (1) 

64 r 3 n {1) 

Equation (1) may be used for finding the load corresponding 
to an assigned deflection in a given spring. The equation can 
be put in the following form for finding the deflection due to a 
given load : 

-- 64 P r* n 



d 4 E. 



(2) 



Or the equation may be employed for designing a spring in which 
the load and deflection are given, by assuming any two of the 
three quantities, r, d and n. The most convenient form for this 
latter purpose is usually, 

n = 6^P? (3) 

These equations for rigidity hold good only within the elastic 
limit of the material, as E s is simply a ratio between stress and 
strain within this limit. It therefore becomes necessary to check 
any of the above indicated computations for strength, and it will 
often be found, after thus checking, that the stress is either too 
high for safety, or too low for economy. 

The formula for the strength of a solid circular-section rod 
under torsion is 



- 



SPRINGS 131 



p r = — P d* .\ P = 

16 16 r 



*pd 3 i6Pr 

r = ~i6~P ; P = ~~^d*~ 



(4) 



It is to be remembered that as equation (4) is for safe strength, 
the load (P) should be the maximum load to which the spring 
can be subjected; but equation (3) may be used with any load 
and the corresponding deflection. 

Example: The load on a helical spring is 1600 lbs., and the 
corresponding deflection is to be 4". Transverse modulus of 
elasticity of material = 11,000,000, and the maximum intensity 
of safe torsional stress = 60,000 lbs., wire of circular section. 
To design the spring, assume d = y&", and r = i>£"; from eq. (3), 

4 X 625 X 11,000,000 X 8 

n = 7- — - = 19.4. 

4,096 X 64 X 1,600 X 27 

Checking for the stress by the last equation in group (4), 

16 X 1,600 X 1.5 X 512 
p = — = so, 200 lbs. 

rr X 125 

This stress is found to be safe, but is considerably below the 
limit assigned, and it may be desirable to work up to a somewhat 
higher stress. Another computation can be made (with a smaller 
d or larger r), and by a series of trials, the desired spring can 
be found. The following order of procedure avoids this element 
of uncertainty. The load being given, assume a diameter of wire 
and value of safe stress, then solve in eq. (4) for the radius of coil. 
Make this radius some convenient dimensions (not exceeding that 
computed if the assumed stress is considered the maximum safe 
value). Next substitute these values of d and r (with those 
given for P, d and £,) in eq. (3) to find the number of coils. Thus, 
with the data of the preceding example, assuming d = yi"\ 

- pd? - x 60,000 X 125 
16 P 16 X 1,600 x 512 

If the yi" rod is wound on an arbor 3" diameter, the radius to the 
centre of coils will be about 1.81"; and the corresponding stress 



132 MACHINE DESIGN 

would be 60,500 lbs. per square inch. This is. so slightly in ex- 
cess of the assigned value that it may be permitted, especially as 
this value is a moderate one for spring steel. Substituting in 

eq- (3). 

3 d* E s 4 X 11,000,000 X 625 

64 P r 3 64 x 1,600 X 5-93 X 4,096 

It may be desirable to fix upon the radius of coil, rather than the 
diameter of wire, in the first computation, in designing a spring. 
From eq. (4) : 

16 Pr 



d 3 = 



= L7^ ... (5) 



<P = -^r,.:<l = , 2 6^m ... (6) 



Tip V P 

In other cases, it may be desirable to assume the ratio of the 
radius of coil to the diameter of wire, then from eq. (4) : 

i6Pr 
-pd ' \ p\d 

In either of the preceding conditions, a standard size of wire 
should be chosen. 

In checking a given spring, it may be required to determine 
either the safe load, or the safe deflection. If the former is the 
case, eq. (4) may be used directly. If it is required to find the 
safe deflection, substitute the value of P from eq. (4) in eq. (2) 
and the result is 

12.57 wr 2 ^ / x 

S= EJ (7) 

The weight of a spring is a matter of some importance, as the 
material is expensive. The following discussion shows that the 
weight varies directly as the product of the load and the deflec- 
tion, inversely as the square of the intensity of stress in the wire, 
and directly as the transverse modulus of elasticity. Hence for 
a given load and deflection, economy calls for a high working 
stress and a low modulus of elasticity. From eq. (4) : 

* d? , 
p = — p — • also for a member under torsion, 
16 r 

p = — X — r- 1 [Church's " Mechanics," p. 235]. 



SPRINGS 133 

d i E s d$E s 

.-. p = — x — x — L — = - — H- 

2 r 2 * r n 4 n r n 
a~ r 2 n p 

■■■ 5 - ± Te ± (8) 

". - 2 d 2 rnp 2 • • 

■ vp -'r-?r • (9) 

But the volume of the spring is 

v = J4*d 2 l = 1 / 4 7z2 d 2 m (10) 

i> 2 v 2 £, 

The weight is directly proportional to the volume; hence for 
given values of E s and p, the weight varies simply as the product 
of the load and the deflection. All possible helical springs (of 
similar section of wire) have the same weight for a given load 
and deflection, if of the same material and worked to the same 
stress. It can be shown that a helical spring of square wire must 
have 50 per cent greater volume than one of round wire, the 
stress and modulus of elasticity being the same in both. The 
round section is generally admitted to be best for helical springs 
under ordinary conditions. 

A small wire of any given steel usually has a higher elastic 
limit than a larger one, while there is not a corresponding change 
in the modulus of elasticity with change in diameter. This 
suggests the use of as light a wire as is consistent with other 
requirements. 

An extensive set of tests of springs, conducted by Mr. E. T. 
Adams, in the Sibley College Laboratories, indicates that the steel 
such as is used in governor springs may be subjected to stress 
varying from about 60,000 lbs. per square inch with ^" wire to 
80,000 lbs. per square inch (or more) in wire Y%" diameter. The 
following expression may be used to find the safe stress in such 
springs : 

, 15,000 . 

p = 40,000 + — 2 — ( I2 ) 

Mr. J. W. Cloud presented a most valuable paper on Helical 
Springs before the Am. Society of Mechanical Engineers (Trans., 



134 



MACHINE DESIGN 



Vol. V, page 173), in which he shows that for rods used in rail- 
way springs {%" to i T V diam.), the stress may be as high as 
80,000 lbs. per square inch, and that the transverse modulus of 
elasticity is about 12,600,000. 

Two or more helical springs are often used in a concentric nest 
(the smaller inside the larger) ; all being subjected to the same 
deflection. This is common practice in railway trucks, where 
the springs are under compression when loaded. If these springs 
have the same "free" height (when not loaded), and if they 
are of equal height when closed down "solid," Mr. Cloud shows 
that the length of wire should be the same in each spring of the 
set for equal intensity of stress. The "solid" height of a spring 
is H = dn, and the length of wire is 1 = 2 - r n; hence the num- 
bers of coils of the separate springs of the set are inversely as the 
diameters of the wire and inversely as the radii of the coils; or 
the ratio of r to d is the same in each spring of the nest. This 
conclusion may be somewhat modified when it is remembered 
that the wire of smaller diameter may usually be subjected to 
somewhat higher working stress than the larger wire of the outer 
helices; and also that the wire of these compression springs is 
commonly flattened at the end to secure a better bearing against 
the seats. See Fig. 30. 



Pr 



16 



pd 



SUMMARY OF HELICAL SPRING FORMULAE. 

12.57 n r 2 p 



pd" 



r = 


16 P 


d = 




d = 


-Wf© 


/> = 


i6Pr 

nd* ' ' 


P - 


npd 3 



16 r 



(i) 
(ii) 

cm) 

(IV) 

(V) 

(VI) 



s 






n 



v 



EJ 


^ d 4 E s 


64 r 3 n 


64 Pr*n 


d'E, ■ 


WE. 


64 Pr 3 ' 


2E - PS 



(VII) 
(VIII) 

(IX) 

(X) 

(XI) 



SPRINGS 135 

Two common methods of attaching "pull" springs are shown 
in Fig. 30 (a) . One end of the spring shows a plug with a screw 
thread to fit the wire of the spring. This plug is usually tapered 
slightly, and the coils of the spring are somewhat enlarged by 
screwing it in. The other end of the spring shows the wire bent 
inward to a hook which lies along the axis of the helix. The 
former method is usually preferable for heavy springs. 

Formulae (I) to (VII), inclusive, relate to strength; (VIII) 
to (X), inclusive, relate to rigidity, or elasticity. 

In the absence of more exact information as to the properties 
of the material of which a steel helical spring is made, the fol- 
lowing values may be taken: 

E a = 12,000,000, 

15,000 

p = 40,000 + : . 

a 

44. Spiral or Helical Springs in Torsion. The following 
formulae for either true spiral or helical springs subjected to tor- 
sion are derived from "The Constructor," by Professor Reu- 
leaux. 

PRl PR 

9 ~~ EI ' P ~ Z ' 
In which 

P=load applied to rotate axle, 
R = lever arm of this load, 
<p = angle through which axle turns, 
/= length of effective coils, 
£ = modulus of elasticity (direct), 
/ = moment of inertia of the section. 



>p 



CHAPTER VI 
RIVETED FASTENINGS 

45. General Considerations. The simplest form of fasten- 
ing is the rivet. It consists of a head a (Fig. 31), a straight shank 
b, and a second head c, which is formed while hot and known 
as a point. When it is desired to rivet two pieces M N to- 
gether, mating holes are punched or drilled as shown, the rivet 
is heated white hot and pushed into the hole which is purposely 
made a little larger in diameter. The head is held up firmly 
against the plate by a heavy bar or sledge and the point may be 
formed with a hand hammer, or with the aid of a forming tool 

or set. In riveting on a large 
scale this operation is per- 
formed by hydraulic or pneu- 
matic machines. The relative 
merits of the two methods will 
be more apparent after further 
discussion. The rivet is a per- 
manent fastening and cannot 
be removed without the de- 
struction of either head or point. 
It is largely used in structures 
such as bridges, the framing of buildings, ship work, boilers, 
tanks, etc. 

Fig. 32 shows various forms of rivet heads and points. The 
form shown at B is most commonly used for small rivets up to 
T 5 g in. diameter, which are driven without heating, for such work 
as light tank and smokestack work. The form at C is much 
used in ship work, or wherever smooth exterior surfaces are 
desired. In machine work, where great accuracy is required, 
the holes are reamed, and the rivet carefully fitted so as com- 
pletely to fill the hole; both heads in such cases are usually 
countersunk and formed cold. 

1^6 




Fig. 31. 



RIVETED FASTENINGS 



137 



When a rivet is "driven" hot it shrinks in cooling, drawing 
the riveted parts firmly together. When cold it is under a ten- 
sile stress due to this shrinking, and for the same reason it is 
always a little smaller than the hole which it originally com- 
pletely filled when hot. The tensile stress due to this cooling 
effect cannot be accurately determined as it depends on the tem- 
perature of the rivet, and the manner in which it is driven. Rivets 
are, for this reason, unreliable as tension members and are seldom 
so used. In most cases the parts M and N (Fig. 31) have the load 
P applied as shown, and the tendency is to shear off the rivet 
and produce relative sliding between M and N. The normal 
load, P', due to the tensile stress in the rivet, holding the surfaces 
of M and N firmly in contact, sets up a frictional resistance equal 
to nP' which opposes the action of P. From experiments made 



Conical 




Fig. 32. 



by Stoney * it appears that this frictional resistance may be taken 
at about 10,000 lbs. per square inch of rivet area. Experiments 
by Bach, and others, show a much higher resistance, but it is 
evident that if the normal pressure of the rivet is such that a 
stress equal to or greater than the elastic limit is induced, the 
permancy of the resistance cannot be relied upon. 

In some French and German practice the design of the joint is 
based entirely upon the frictional resistance, but in England 
and America it is neglected, and the design based upon the 
tensile and shearing strength of the plates and rivets. 

46. Forms of Joints. Riveted joints are of many forms de- 
pending on the character of the work to which they are applied. 
In structural work, such as bridges, they are used simply to 



* " Strength of Riveted Joints," page 75. 



I38 MACHINE DESIGN 

resist direct loads; but in boiler construction, and similar work, 
they must not only resist direct loading but must also be tight 
against fluid pressure. This last requirement materially affects 
the proportions of the joint, and makes the design of joints for 
withstanding fluid pressure most important. Riveted joints are 
divided into two general forms. 

(a) Lap Joints, where the sheets to be joined are lapped on 
each other and riveted as shown in Fig. 33 (a). 

(b) Butt Joints, where the edges of the sheets to be joined 
abut against each other, and have auxiliary butt straps or cover 
plates riveted to the edge of each, as shown in Fig. 33 (e) and (f). 

A lap joint may have one or more rows or seams of rivets, 
and these rows way be arranged in the form of "chain" rivet- 
ing, Fig. 33 (b), or in the form of zigzag or staggered riveting, 

Fig. 33 (c). 

A butt joint may have one or more seams of rivets on each 
side of the joint, and these may also be arranged in either chain 
or staggered form, as shown in Fig. 33 (f) and (g). 

The combinations that may be thus made up are very numer- 
ous, and the student is referred to any treatise on boiler work for 
fuller information on this point. 

The distance between rivets along the seam is called the pitch 
or spacing, and will be denoted by s, Fig. 33 (b) . An examina- 
tion of any riveted joint will show that the arrangement of 
rivets, or pattern as it may be called, continually repeats itself as 
the "seam" extends along the joint, the repetition occurring with 
the greatest pitch, where the pitch of the various seams is unequal 
as in Fig. 33 (h) . A unit strip is equal in width to the pitch, the 
maximum pitch being taken when the pitch of all seams is not 
the same. The transverse pitch is the distance between the 
centre lines of adjacent seams Fig. 33 (b) and will be denoted 
by s t . The diagonal pitch is the distance between the centre of 
a rivet and that of the one nearest to it diagonally, in the next 
row, and will be denoted by s d Fig. 33 (d). The margin is the 
distance from the edge of the plate to the center line of the nearest 
row of rivets, as e Fig. 33 (d). It is sometimes defined as the 
distance from the edge of the plate to the edge of the rivet hole. 



RIVETED FASTENINGS 



139 



47. Stresses in Riveted Joints. The stresses that exist in 
the various members of riveted joints are complex, and do not 
admit of refined calculation. Not only are the plates subjected 
to the apparent direct stresses of tension and compression, and 



6 



&=l 






St- •+«■■- 



■>|'5r-]<- 



<E^ 
^^> 



/\ A 







(<0 










/\ /\ /\ / \ 
U^ vl|V sip si/ 

(8) 



^> 



<B 






v 

-cb- 



e 



z:^ 



^7 



-e- 



-e 



<^ 



<^ 




.m 



4^V 



X[7 vO/ 



-T 



-l-M • 1 1--, J— 4- -A 



^ 



w 



\j^ kw 



Fig. 33. 
(a, b, c, d, e, f, g, h.) 



A- 



140 



MACHINE DESIGN 



the rivets to shear and compression, but often there are also 
bending actions which are difficult to analyze and provide for 
mathematically. Thus a simple lap joint, as that shown in Fig. 
33 (a), when subjected to a load, tends to take the form shown 
in Fig. 34. The force applied tends to draw the plates into the 
same plane, putting a bending action on the plate and rivet, a 
greater tensile stress on the rivet head, and a concentrated crushing 
load on the corners of the sheets. The frictional resistance is 
entirely destroyed when the conditions illustrated in Fig. 34 exist. 

The above defects are more marked in the lap joint than in 
the double strapped butt joint, as in the latter the plates are ini- 



tio. 34. 



Ca) 



OO 



(b) 




tially in line and the condition shown in Fig. 34 cannot occur. But 
even here the rivets do not completely fill the holes when cold, and 
hence some bending of the rivet and concentrated crushing on the 
plate must result. Again, while the quality of the material 
forming the joint may be well known or determined, the work- 
manship is not so easily controlled and may be very defective 
and yet not show on the exterior; and while there have been 
many tests * made to find the ultimate strength of riveted joints, 

*Proceedings Institute of Mechanical Engineers, 1881, 1882, 1885, 1 
Watertown Arsenal Reports, 1885, 1886, 1887, 1891, 1895, 1896. 



RIVETED FASTENINGS 141 

such tests show only the stress at which a certain element of the 
joint failed, and do not throw any light on the distribution and 
progress of the stresses in the various individual members dur- 
ing the test. Such tests have usually been performed on joints 
made of straight plates while in practice these are often curved. 
These experiments, therefore, while giving the only data available 
relative to the ultimate strength, should be used with judgment 
in designing. For these reasons the theoretical formulae deduced 
for the design of riveted joints, as a rule, take cognizance only 
of the apparent simple stresses and provide for the unknown by 
means of a factor of safety. 

It has been found that riveted joints may fail in one of the 
following ways : 

(a) Shearing of the rivet as in Fig. 35 (a). 

(b) Rupturing of the plate by tension as in Fig. 35 (b). 

(c) Tearing of the margin as in Fig. 35 (c). 

(d) Shearing of the margin as in Fig. 35 (e). 

(e) Crushing the plate, or rivet as in Fig. 35 (d). 

(f) Rupturing of the plate diagonally between rivet holes by 
tension, in staggered riveting. 

Where the joint is complex in form, ultimate failure may be 
due to one or more of the above causes. The Watertown Arsenal 
reports include cuts of ruptured joints which are very instructive 
on this point. Figures (a) and (b) , on Plate I, are reproduced 
from these reports, and show very clearly all the ways in which 
failure may occur in the plate. Fig. (c) shows a rivet that has 
been tested to destruction in single shear, while Fig. (d) shows 
one that has been similarly tested in double shear. 

It is obvious that no riveted joint can be as strong as the un- 
perforated plate, since the very fact of making holes in it reduces 
the cross-sectional area in the line of the rivet holes. The ratio 
of the strength of the weakest element of the joint, to the strength 
of the unperforated plate, is called the relative strength or ef- 
ficiency of the joint. The first expression is more suggestive 
and will be used in this work. It is desirable to reduce the 
strength of the plates as little as possible by perforation; and if, 




(a; 





<c) Cd) 

*4 2 Plate I. 




RIVETED FASTENINGS 1 43 

therefore, the correct relation between the size of rivet and cross 
section of perforated plate, for equal strength, is established, an 
excess of strength in other directions, as marginal distance, is 
not a defect but good design, as it insures that the full strength 
of the perforated plate will be in service before rupture can occur. 
A well-designed joint should hence fail by tearing of the sheet 
along the line of the rivet holes, at about the same load as will 
destroy the rivets; and the relative strength of a well-designed 
joint should be the ratio of the cross section of the perforated to 
that of the unperforated plate, the shearing and crushing resist- 
ance of the rivets being equal to the former. If this equality does 
not exist, the relative strength of the joint can be made greater 
by strengthening the weaker of these elements at the expense of 
the stronger. 

48. Marginal Strength. The width of margin is independ- 
ent of the proportions of the other elements, and hence can be 
made sufficient to prevent tearing or shearing, as in Fig. 35 (c) 
and (e). It has been found that, with the usual proportions, if 
the margin be made equal to one and one half the diameter of 
the rivet, it will be safe against both shearing and tearing from 
rivet pressure. A Committee of the Master Steam Boiler Makers 
Association recently recommended, as a result of experiments, 
that the distance from the center of the rivet to the edge of the 
plate be made twice the diameter of the rivet, in order to insure 
excess strength enough to either shear the rivets or rupture the 
plate by tension. It is important, however, that the margin be not 
excessive in boiler work as this makes it more difficult to make a 
steam-tight joint by calking the edge of the plate. The con- 
sideration of the marginal strength can hence be omitted as far 
as its influence on the relative strength of the joint is concerned, 
as it can be made to depend on the diameter of the rivet. 

49. Transverse and Diagonal Pitch. It has been deter- 
mined, in a similar way as above, that in chain riveting the 
transverse pitch should not be less than twice the diameter of the 
rivet or 2d, where d is the diameter of the rivet, and 2.5 d is 
better. It has also been demonstrated mathematically (see 
Cathcart's " Machine Design," page 148) that in staggered riveting 



144 MACHINE DESIGN 

the transverse pitch should not be less than 0.4 times the pitch 
along the seam, in order to avoid rupture along the diagonal 
pitch, and a greater distance is recommended for safety. Unwin 
(page 123) gives 2 d as the minimum diagonal pitch in staggered 
riveting, which would make the transverse pitch 1.7 d, and 
recommends that somewhat greater distances be used for added 
strength. An examination of the practice of several boiler-mak- 
ing and insurance concerns shows that these values check fairly 
well with practice. It appears from the above that the trans- 
verse pitch can also be made to depend on the diameter of the 
rivet, though it is not a direct function of the rivet diameter. 

50. Theoretical Strength of Riveted Joints. Since the margin 
and transverse pitch can be assigned from the diameter of the 
rivet, three of the ways in which a joint may fail, namely c, d and 
/, page 141, can be omitted from the theoretical discussion of the 
strength of riveted joints, leaving a, b and e to be considered; 
the problem being so to proportion the rivet and the pitch along 
the seam as to give equal strength against failure in any of these 
three ways. Let, — 

t— thickness of plate in inches. 

d = diameter of rivet in inches after driving. 

5 = pitch of rivets in inches. 

p t = tensile strength of plates in pounds per sq. inch. 

p t ,= crushing strength of plates or rivets in pounds per sq. 

inch, if rivets are in single shear. 
p' c = crushing strength of plates or rivets in pounds per 

inch, where rivets are in double shear. 
p s = shearing strength of rivets in pounds per sq. inch, 

when in single shear. 
p\ = shearing strength of rivets in pounds per sq. inch, 
when in double shear. 
It is known that the unit shearing resistance of a rivet is 
greater in single shear than in double shear, while the unit 
crushing resistance is less in single shear than in double shear. 
Consider first a simple lap joint (see Fig. 33 a) . The tensile 
strength of the unperforated strip is 

P = stp, .. .. ' (1) 



RIVETED FASTENINGS 145 

The tensile strength of the perforated strip along the seam of 
rivets is 

T = (s-d)tp t (2) 

In the simple lap joint there is but one rivet per unit strip and 
its shearing strength is, 

xd 2 

S = —P> (3) 

4 

The resistance to crushing of the rivet or the plate against which 
it bears is, 

C = dtp* (4) 

For uniform strength against rupture and hence for greatest 
relative strength, 

T = S =C 
Equating (3) and (4) 

*d 2 

4 

,\ d = 1.27/^ 
/ A 
Equating (2) and (3) 

*d 2 
\s-d)tp % = -—p a 
4 

. .7854^ A , 
. . s = + a 

IP* 

mi 1 • 1 T (s — d) t p t s — d 

The relative strength = — = — — — = 

6 P stp t s 

Double-Riveted Lap Joints. In a similar way equations 
may be developed for any other form of joint. Thus for double- 
riveted lap joints 

j 4 P° 

d = 1.27 t £ - 

1.57 (Pp a 
and s = — — — - + d 
^A 

* It is known from experiments on indentation that the resistance to indentation 
depends very little on the form of the indenting body but mainly on its projected 
area. Hence it is customary to take the resistance of rivets to crushing as propor- 
tional to their projected area. 
10 



146 MACHINE DESIGN 

5 d 

The relative strength = and will be greater than in the cast 

of the single-riveted lap joint since s is greater in proportion 

tod. 

Single-Riveted Butt Joints with double cover plates. 
Here, 

d = .64*^7 

P * 

1 • 57 & P's 
s = — — — *- + d 

'A 



and the relative strength = 



s — d 



s 

Double-Riveted Butt Joints with either chain or staggered 
riveting and double cover plate. 
Here, 

s — d 

Note that the relative strength = , and compare the rel- 
ative values of s and d in this case with those in single riveting. 

Thickness of Cover Plates. It is evident that where only 
one cover plate is used its thickness should not be less than that 
of the main plate; and in practice, single cover plates are made a 
little thicker than the main plate to insure an excess of strength. 
A butt joint with a single cover plate is shown at Fig. 3$ (e). A 
joint of this kind is really equivalent to two lap joints. They 
are used where a smooth surface is desired or in such places as 
the longitudinal seams of steam boilers where a lap joint has 
sufficient strength. Double butt straps should not be made less 
than half the thickness of the main plate, and, for the same 
reason as above, it is not unusual to increase their thickness to 
about -fa t where the cover plates are the same width. Where 
the outer cover plate is narrower than the inner plate, as in 



RIVETED FASTENINGS 147 

Fig. 33 (h), the outer cover plate is often of the same thickness as 
the main plate and the inner one from T 7 T to T % /. 

51. General Equations for Riveted Joints. The funda- 
mental equations for riveted joints may be put in a more general 
form. The unit strip, as before, is of width equal to the pitch; 
the maximum pitch being taken for such width of unit strip if 
all rows do not have the same pitch. 

Let k x = p c -f- p s ; k 2 = p' c + p' s ; n = number of rivets per 
unit strip in single shear and w = number of rivets per unit 
strip in double shear per unit strip of joint. 

The general expression for the net tensile strength of the unit 
strip is 

T = (.s-d)tp* (1) 

The general expression for resistance to shearing of the rivets 
in the unit strip is 

n* (P 2 tn* d? , 

S = — — A+ — - — p\ .... (2) 

4 4 

The general expression for resistance to crushing of the rivets 
in the unit strip is 

C = n d t k t p s + m d I k 2 p\ . . . . (3) 

The tensile resistance of the solid strip is 

P = stp t (4) 

Equating 5 and C, eqs. (2) and (3) and solving for d 

A nKp. + mKp: __ () 

n p a + 2 m p: °' 

which gives the proper diameter of rivets for a given thickness of 
plate, when the number of rivets in single shear and the number 
in double shear and the corresponding shearing and crushing 
resistances are known. 

* Where the rows of rivets do not all have the same pitch, as in some forms of 
butt joints, the outer row or that farthest from the edge of the sheet has the greatest 
pitch (see Fig. 33 h). It is evident that if the sheet yield at all by tearing, it will 
yield along this outer row of rivets; for it cannot tear along an inner row without 
shearing the outer row of rivets, and it cannot shear one row of rivets without shear- 
ing all, in which case the joint would yield by shearing of the rivets and not by 
tearing. 



148 MACHINE DESIGN 

Equating T and S, eqs. (1) and (2), and solving for 5 

s = 4 d \ 7J t ) +d ' ' ■ ■ (6) 

Equating S and C, eqs. (2) and (3) 

*d 2 

— - (» p a + 2 m #/) = dt{nk x p % + mk 2 p') 

4 

b , *(P(np a +2tn p/) 

. . dt = — ; — ; : r .... (7) 

4(»^iA + W ^A0 v " 

Equating T and 5, eqs. (1) and (2) and solving 

stp t = — {np s +2tnp:)+dtp t =P. . . (8) 
4 

If the joint is designed for maximum relative strength, T = 
S=C, hence any one of these three quantities divided by P 
gives the relative strength (£) of the ideal joint, for any given 
form, or dividing (2) by (8) 

—(np M + 2mpl) 
E =- = 



P ~ *d? 

(n p H + 2 m p s ') + dtp, 

4 

Substituting the value of d t as given by eq. (7) and dividing 

*d 2 

numerator and denominator by (n p 8 + 2 m p a '), 

4 

- ,+ v. ■ ■:•'•• » 

nk ± p a -\- m k 2 pi 
If all rivets are in single shear, 

If all the rivets are in double shear, 

E- 1 -t- • (9") 

mk 2 p , 



RIVETED FASTENINGS 149 

Equation (9) applies to any form of riveted joint. It is 
useful in finding the limiting relative strength of joint for any 
form and materials; the actual proportions adopted may give a 
lower relative strength, but can never give, higher relative strength. 

These general equations were originally due to Professor 
William N. Barnard, who also suggested the above expressions 
for the maximum relative strength in the general case. 

The forms in which eqs. (9), (9') and (9)" are now given are 
due to Professor H. F. Moore. 

The following are rough average values of the relative strength 
of joints as made in practice for boiler work: 



Single riveted lap joints 

Double riveted lap joints .... 
Single riveted butt joints 
Double riveted butt joints 
Triple riveted butt joints .... 
Quadruple riveted butt joints 



55 
70 

65 

75 
80 

85 



52. Practical Considerations Affecting Proportions of Riveted 
Joints. It is to be especially noted that the proportions of 
riveted joints as given by the foregoing equations are based 
on equal strength of rivet and plate, and that any variation 
therefrom will destroy this theoretical equality. It is apparent 
also that any variation in the strength of the material used would 
affect the proportions as given by these equations, and that a 
table of rivet diameters and pitches would have to be very ex- 
tensive .to cover the entire range of practice. It is an advantage 
in practice, however, to adopt regular diameters and pitches for a 
given thickness of plate and form of joint. It has also been 
found that as the thickness of the plate increases, the correspond- 
ing theoretical diameter of the rivet sometimes becomes too large 
to be easily driven, especially in the case of simple lap joints. 
In the case of boilers, or wherever fluid pressure must be with- 
stood, the theoretical spacing must sometimes be modified in 
order that it may not be so great as to prevent the making of a 
steam-tight joint. Wherever such variations are made the general 
expressions for T, S and C can always be used to check the 
strength and show in what direction the joint may be strength- 



150 MACHINE DESIGN 

ened, with the fundamental object of making it strong enough 
in all other directions to insure full service out of the plate itself. 
It may be noted that the bearing resistance of a rivet varies with 
the diameter, while the shearing resistance varies with the square 
of the diameter. If, therefore, the rivet chosen be smaller in 
diameter than would be given by the theoretical equations, the 
shearing resistance alone need be regarded; while if the diameter 
of the rivet be greater than the theoretical diameter, the bearing 
pressure only need be considered. 

If the joint to be made does not have to withstand fluid or 
gaseous pressure, the design can, for ordinary thickness of 
plate, be made to conform closely to the theoretical proportions 
for equal strength; but when fluid or gaseous pressure must be 
withstood, as in boiler work, the spacing of the rivets for thick 
plates must be less than the theoretical spacing to insure tight- 
ness; and in all cases as the plates increase in thickness, the 
diameter of the rivet, as already noted, is for practical reasons 
reduced in diameter from that required for equal strength. The 
relation between diameter of rivet and thickness of plate as fixed 
by average practice may be expressed by the equation, 

d = r.2 v 7 / (10) 

For plates above y&" thick this equation will give rivets smaller 
in diameter than required for equal strength in all directions. 
As before pointed out, the rivets in such cases need only be 
checked for shearing strength. If the diameter of the rivet be 
determined by equation (10), and the pitch so chosen as. to make 
the tensile strength of the perforated plate equal to the shearing 
strength of the rivet, the maximum relative strength of joint 
possible with the rivet chosen will be obtained, and it will be 
found that the joints will be steam-tight. 

Example. It is required to design a riveted joint, as shown 
in Fig. 33 (h), for a boiler shell in which the force tending to pull 
the joint apart is 6,000 pounds per inch of length of shell. The 
plate is to be of steel of 60,000 lbs. tensile strength, the rivets 
are to be of steel and are to have a shearing strength of 49,000 
lbs. per square inch in single shear, and 42,000 lbs. per square 
inch in double shear. The factor of safety is to be 5. 



RIVETED FASTENINGS 151 

The allowable stress per square inch of the shell outside 

... . 60,000 , . 

the joint is = 12,000 lbs. If the joint were as strong as 

the unperforated plate the thickness of the plate would be 

6,000 

= X inch. The relative strength of the joint will not 

12,000 ° J 

be less than .80, and hence the thickness of the plate must be 

# -V 

.80 8 ' 

The diameter of the rivets will = 1.2 \| — = — nearly. 

^8 8 

The size 0} the punched hole and the diameter of the driven 
rivet will be ff". Equation (6) of this chapter gives the rela- 
tion between pitch and diameter of rivet for equal strength 
against shearing of the rivets and tearing of the plate. 

4 L tp x J ^ 

Here, n = 1 and m = 4. 

Hence, * = lMl[ *9,°°o + (2 x 4 * 42,ooo) -J tf 
4 L $4 X 60,000 J * 6 

.".5 = 8 inches nearly. 

The relative strength of the joint is 



8 



15 
T6" 



= 88%, hence the design is safe. 



8 

If the pitch found as above should be considered too great, 
either on account of very high steam pressure or because it is 
desired to make the structure stiffer, a smaller pitch could be 
used, but the relative strength would be less. 

Where no fluid pressure is to be withstood the above methods 
will always give satisfactory results for joints in tension. For 
joints in compression the student is referred to treatises on struc- 
tural work. 

52.1. The Making of Riveted Joints. It is evident that the 
following precautions must be observed in making first-class 
riveted joints. 



I52 MACHINE DESIGN 

(a) The plates must be in close contact before the rivet is 
driven, to prevent a fin from forming between them and thus 
making a tight joint impossible. 

(b) The mating holes must be "fair"; that is, they must be in 
perfect alignment to insure full cross section of the rivet at the 
junction of the plates. 

(c) The rivet must completely fill the hole. 

(d) The rivet should be carefully driven so that its strength, 
or that of the plate, will not be weakened by poor workmanship. 

(a) In hand riveting the plates are drawn up together, before 
the rivet is driven, by a bolt placed in a hole near that in which 
the rivet is to be driven. With comparatively thin plates this 
method will accomplish the result if the holes have been accu- 
rately spaced, and if the plates have been properly rolled and fit 
well. For heavy work, where for other reasons machine riveting 
is necessary, the riveter is sometimes provided with a power- 
driven closing device which holds the plates up till the rivet nips 
from cooling. 

(b) The rivet holes in the plate may be either punched or 
drilled. They are generally made about iV inch larger than the 
rivet. Generally speaking it is cheaper to punch the holes than 
to drill them, and hence in the cheaper kinds of work, and with 
thin plates, punching is almost always resorted to. In structural 
work the holes are generally punched. There are, however, some 
serious objections to punching. When the punch is forced 
through a plate the amount of metal which it removes in the form 
of a "plug" is not equal to the amount that originally filled the 
hole. This is accounted for by the fact that punching is not a 
pure shearing action, but that during the process there is a flow 
of metal from under the punch to the walls of the hole, setting 
up a stress in the material, and, if the metal is at all hard, seri- 
ously impairing the strength of the plate. It is found that this 
action is confined to a thin ring next to the hole, and that by 
either reaming out the hole about tV inch all around, or anneal- 
ing the plate, this weakening effect disappears. The process of 
punching is apt to make inaccurate work and, therefore, when 
the plates are brought together the mating holes are not fair. The 



RIVETED FASTENINGS 1 53 

old practice of driving a taper drift pin into such holes and 
drawing them into line by force is now largely prohibited, 
the injury thus done to the plate being often very serious. If, 
however, the holes are punched a little small, and put together 
and reamed to the proper size, the difficulties due to punching 
are largely overcome. Thin plates, in which the effect of punch- 
ing is small, are punched and used without reaming or annealing. 
Plates more than y 2 inch thick should always be either annealed 
or have the holes reamed after punching. Heavy plates are 
always better if drilled, and all first-class boiler work requires 
the holes to be drilled in place and all burrs carefully removed. 
This last is important, as the burr, if allowed to remain, may 
seriously impair the strength of the head. A small countersink, 
on the other hand, materially contributes to the strength of the 
rivet. When plates are annealed the work should be properly 
done ; for if the plate be overheated structural changes take place 
that materially weaken the metal. The heating should not be 
too rapid nor the temperature above a medium cherry red. The 
holes made by punching are necessarily somewhat tapering in 
form, and where they are used as they come from the punching 
machine the rivet holes should be punched so that they will come 
together as in Fig. 36; for the rivet drives better and the tapering 
rivet has a tendency to relieve the head of part of the tensile stress. 

(c) Since it is necessary to have the hole a little larger than 
the rivet, it is clear that the rivet when driven must be upset 
throughout its entire length in order that it may completely fill 
the hole. Large rivets should therefore be machine driven, as it is 
difficult to upset heavy rivets, especially if of great length, by 
hand. If a rivet does not completely fill the hole an undue con- 
centrated bearing, or shearing stress, may be brought on its 
neighbor. 

(d) On the other hand, care must be exercised in machine 
riveting that the pressure applied does not create such a flow in 
the rivet as unduly to strain the plate, and also that the pres- 
sure applied is not great enough to crush the plate directly. 
Practice allows about 80 tons per square inch of rivet area. 
Machine riveting, when well done, is superior to hand work, 



154 MACHINE DESIGN 

the plates being held up firmer, and also because the impact 
from hand riveting, especially if the rivet is worked too cold, is 
liable to result in the breaking off of the head. In either case 
care should be exercised that the point is formed on the rivet 
concentrically; if the dies are not properly set eccentric rivet- 
ing will occur. In machine riveting the pressure should be re- 
tained on the rivet till cold enough to hold the plate firmly. 
This is sometimes recognized in writing specifications. 

If the spacing of the rivet is correct and the riveting well 
done, the joint will be tight against ordinary pressures. Where 
tight joints do not result the edge of the plate is " calked." This 
is often done, as shown in Fig. 36, by means of a sharp-nosed 
tool T which tucks the sharp bevelled edge of the plate under- 
neath, as shown in an exaggerated manner. There is liability 
of injuring the lower plate in using a sharp-nosed tool as T, and 
the method shown at B, Fig. 36, is preferable. The plates should 
be bevelled before riveting, as the method of hand bevelling after 
riveting, as often done in practice, is almost sure to result in some 
injury to the lower plate. 

52.2. Strength of Materials for Riveted Joints. It is well 
known that the strength of rivets is different in single and double 
shear. The following may be used as average values. 

p s* Pcf 

Iron rivets single shear 40,000 60,000 

Iron " double " 39,000 72,000 

Steel " single " 49,000 80,000 

Steel " double " 42,000 100,000 

Steel Plates for Boiler Work are generally specified to have 
a tensile strength of not less than 55,000 lbs. per square inch, 
and not more than 65,000 lbs. per square inch, for if the tensile 
strength is too high and the metal is hard they are liable to 
crack while being worked. For structural steel construction 
the student is referred to handbooks on structural work. For 
iron plates an average value may be taken as 45,000 pounds per 

* Master Boiler Makers' Association Rules, page 150. 

t Proceedings Inst. Mech. Engineers, 1885, and Unwin's "Machine De- 
sign," page 132. 



RIVETED FASTENINGS 1 55 

square inch. It is shown by experiment that the metal between 
the rivet holes has a higher apparent tensile strength than that 
of the unperf orated plate; this increase being sometimes as 
high as 20%. It is questionable, however, if this should be 
taken account of in designing, especially where the holes are 
punched, as the operation of punching may more than offset 
this peculiar increase. 

52.3. Factor of Safety. In boiler work the factor of safety is 
taken at about 5 which, of course, brings the working stress well 
below the elastic limit. If the joint is to be subjected to hydraulic 
pressures, where heavy shocks may have to be withstood, this 
factor should be increased. 

52.4. Practical Rules. It has already been noted that practi- 
cal considerations make it necessary to modify the theoretical 
equations for uniform strength. There are many sets of practical 
rules for designing riveted joints a number of which will be 
found in the references given below. 

Rules of the Hartford Steam Boiler and Inspection Co. 

Rules of the American Bureau of Shipping. 

Rules of the Master Steam Boiler Makers' Association. 

Rules of the U. S. Board of Supervising Inspectors. 

See also Cathcart's "Machine Design." 

Proceedings of Inst, of Mech. Engineers, 1885. 

Unwin's " Machine Design." 

Wm. M. Barr's " Boilers and Furnaces." 

" Steam Boiler Construction," W. S. Hutton. 



CHAPTER VII 

SCREWS AND SCREW FASTENINGS 

53. Form of Screws. Screws, as used in machines, may be 
divided into two classes. 

(a) Screw fastenings. 

(b) Screws for transmitting power. 

The form of the thread depends upon the service required. 
Thus, for screw fastenings, the full V as shown in Fig. 37 (a), or 
modified forms of V threads, as shown in Fig. 37 (b) and (c), are 
most used because they are strong and easily cut by machine 
dies. They are inefficient for transmitting power, but this is a 
desirable quality in fastenings, as it reduces the liability of un- 
screwing. For transmitting power the square thread, Fig. 37 (d), is 
most used, since its efficiency is higher than that of any other 
form. It cannot be cut with a die, however, and it is difficult to 
compensate for wear with this form of thread. For these reasons 
the half V thread, Fig. 37 (e), is often used for transmitting power 
when wear is an important factor. In Fig. 37 (e) the Acme 
standard thread of this form is shown. The efficiency of this 
form of thread is a little less than the square thread but it can be 
cut with a die and wear can be compensated for by means of a 
longitudinally split nut; this compensation making it very de- 
sirable for such service as lead screws of lathes, etc. Fig. 37 (f) 
illustrates the buttress thread, which is often used to exert pressure 
in one direction only. The pressure face is perpendicular to 
the axis of the screw, and the back face usually makes an angle 
of 45 with this axis. This screw has, therefore, the efficiency 
of the square thread and the strength of the V thread. The 
underlying principles of all screws are the same, and before dis- 
cussing the various forms and classes in detail the fundamental 
equations relating to their action will be developed. 

1^6 



SCREWS AND SCREW FASTENINGS 



157 



54. Friction and Efficiency of Square Threaded Screws. In 

Fig. 38 (b), let AT" represent a nut moving on a square thread, 
under the action of a tangential force P, acting at the mean 
radius of the thread. Let this force P be applied by means of a 
couple, so that there is no lateral pressure against the screw. 
Let W represent the load under which the nut is moved, and 
consider that it can move only in an axial direction, hence there 
is friction between N and W. This frictional force F x (Fig. 38 b) 
may or may not act at the same radius as P, and the work due 
to this frictional force will vary with the radius at which it acts. 
It can be considered as forming a resisting moment opposing the 



c60 




Full V 
(a) 




Sellers or U.S. Standard 
(b) 




Acme Standard Thread 
(e) 



Fig. 37. 



Buttress Thread. 
(f) 



turning moment due to P, hence, when computing the re- 
quired turning moment of P, is to be added to that value. It 
can therefore, for simplicity, be omitted temporarily from the 
discussion. 



Let p 

."1 
d 

r 

d t 



coefficient of friction between thread and nut. 
coefficient of friction between load and collar, 
nominal or external diameter of screw, 
nominal or external radius of screw, 
diameter of screw at bottom of thread, 
radius of screw at bottom of thread. 



158 



MACHINE DESIGN 



r — 



d = 



frictional radius of collar, 
mean diam. of thread = -' 



mean radius of thread = 



2 
r + r x 



pitch, or angular advance of thread per turn, 
angle made by thread with a plane perpendicular to 
the axis of screw. 




Fig. 38(a) 



If now the thread be developed as in Fig. 38 (a), it is seen, 
since the thread is a true helix, that the action of the thread 
and nut is identical with that of a body N sliding up an inclined 
plane of length nd m , and vertical height 5 equal to the pitch, and 
carrying a load W which is free to move vertically only. Omit- 
ting F x temporarily, the forces acting are the load W, the fric- 
tion F 2 between the thread and nut, the driving force P, and the 
normal reaction R. It is required to determine for any angle 
«, the value of P required to slide the body N (turn the nut) 
up the incline. The frictional resistance F 2 = nR. Hence, 
resolving all forces parallel to ac 



SCREWS AND SCREW FASTENINGS 



!59 



P cos « — n R = W sin « . . . . (i) 

n P cos « — W sin a 
.'.* = .... (2) 

Resolving all forces perpendicular to ac 

R — P sin a = W cos « (3) 

Substituting in (3) the value of R obtained in (2) 

P = w r^jL+jL^] . . . . (4) 

L-COS « — j« Sin a-l v ' 

- d s 

Since (Fig. 38 a) cos a = — " and sin a = — } equation (4) 

(Jb C/ (JL is 

may be written 

P = H ,p+JL^] (5) 

The friction, F^^^ W, and if r c be the radius at which F x 
acts, the moment of F x around the axis of the screw = ft t Wr c ; 
and when this resistance is considered the total moment of P 
around the axis is 

Pr m = Wr ri±JLl^l + H Wr . . . (6) 
m L- d — ti s J c v ' 

If the load is being lowered, the directions of F t and F 2 are 
reversed, and in this case the turning moment that must be 
applied is 

p,r m = Wr[ fl7: / m ~ S \ + »Wr c . . . (6') 

In equation 6' the first term of the right-hand side of the 

equation is the moment of the resistance at the thread, while 

the second term is the moment of the collar friction. If 

s 
^7: d m = s, that is if —j- = tan a = p y the moment of the re- 

m 

sistance at the thread will be zero, and if there is no collar 
friction, or if it is very small, as in the case of ball-bearing 
thrusts, this will give a condition of equilibrium, the friction of 
the thread alone just sustaining the load, and P x will be equal to 



l6o . MACHINE DESIGN 

zero. If the pitch s is made greater than jj. * d m , the moment of 
the resistance at the thread becomes negative; and if increased 
till its numerical value is equal to the moment of collar friction, 
the entire right-hand side of the equation will be equal to zero 
and the load will just be sustained by the friction of the thread 
and collar combined. If the pitch is still further increased, the 
entire right-hand side of the equation becomes negative, and 
the moment P x r m must be applied in the direction of raising 
the load, or the screw will " overhaul," the nut exerting a turn- 
ing moment in the downward direction. 

To find the limiting value of « where the nut will not over- 
haul, equate the right-hand side of the equation (6') to zero and 

solve for — — , whence 

* n r m + h r c 

= tan a = .... (7) 



- d r — j"i j« r 

If r =0, in which case there is no moment due to collar fric- 

C ' 

tion, tan « = p- as before. 

Ifr m = r c and/; = ^ 

2 n 
tan a = (8) 

and if // be taken as .1 (see Art. 65) tan « = .2 whence 
a = 11°. 

To find the efficiency of the screw, consider that the load has 
been raised a distance equal to the pitch, that /*!=/*, and r c = r m , 
then 

. work done W s W s 

e = efficiency = r— : = — = — 7—- or 

J energy expended 2* P r m n d m P 

since s = * d m tan «, and inserting the value of P from equation (6) 

W 7zd m tan a 



e = 



L.7T (t m fl s-J 

tan a (1 — pi tan «) . 

= nearly (o) 

tan a + 2 p. J Ky/ 

If the collar friction is zero, or very small, as in the case of 
ball-bearing thrust collars 



SCREWS AND SCREW FASTENINGS l6l 

tan « (i — .a tan «) 

e = (10) 

tan « + /' 

If, in equation (9), /« be taken as .1, and tan " as .2 as before, 
e will equal 50 per cent (nearly), and a brief reflection will show 
that in no case can the efficiency of a self-sustaining hoisting 
screw exceed 50 per cent. Suppose the load (Fig. 38) to be just 
sustained by the frictional resistance of lowering, that is tan a 
= [x or a = angle of repose. If now a force P is ap- 
plied, just sufficient to relieve this frictional resistance, the 
load will be sustained by the force and the reaction R. If 
the frictional resistance of raising were zero, the slightest ad- 
dition to P would move the body up to plane. But the fric- 
tional resistance of raising is equal to that of lowering, and con- 
sequently, before the body can be started up the plane, a force 
2 P must be applied; which is twice the force required to bal- 
ance the frictional resistance, and the efficiency would then 
be 50 per cent. A similar reasoning will apply to other hoisting 
devices which are barely self-sustaining on account of friction, 
namely, that the force which must be applied to start the load is 
equal to the friction due to lowering plus the friction due to 
raising. Hence the maximum efficiency for such self-sustaining 
mechanisms is 50 per cent. 

In designing screws for power transmission it is desirable to 
know the pitch angle that will give maximum efficiency for the 
conditions taken. If the first differential of equation (9) be 
taken and equated to zero, it is found that the maximum efficiency 
when collar friction is considered will occur when 



tan « = \/ 2 + 4 y 2 — 2 ft 

If n = .1 as in transmission screws where lubrication is imper- 
fect, tan a = 1.23 and a = 51 . 

In the case of oil bath lubrication, as in worm gearing, ft may 
be as low as .05 when tan a for maximum efficiency = 1.318 or 

a = 52°-49 / . 

In a similar way from equation (10) for maximum efficiency 
tan « = a/ i + f? — ft 



11 



162 MACHINE DESIGN 

Whence f or fi = .1, a = 42 
and for \i = .05, « = 43 — 34' 

The effect of the pitch angle on the efficiency of the screw is 
of great importance in designing screws for power transmission, 
and is more fully discussed in Arts. 64 and 65. 

55. Friction and Efficiency of Triangular Threaded Screws. 
With triangular threaded screws the normal pressure at the 
threads is greater than with square threads; hence the friction 
at the threads is greater, other things being equal. In Fig. 38 (c) 
the normal pressure for a square thread is indicated by R, while 
the normal pressure for a triangular thread is R'=R sec ?, in 
which <p = half the angle between the adjacent faces of a thread. 
R" represents the radial crushing action on the thread of the 
screw, and its equal and opposite reaction tends to burst the nut. 
With 6o° angular thread, as in the Sellers system, or the common 
V thread, R' =R sec 30° = 1.15^. The friction increases directly 
as the normal pressure; or it is about 15 per cent greater in the 
6o° angular thread than in the square thread. 

If in equations (1) and (s),R sec <p be substituted fori?, then by 
a similar method of reasoning as in square threads, when collar 
friction is neglected, 

• P = w R + ^secq 

L Tt d m — fissecpJ v y 

If the collar friction is considered, the moments around the 
axis when raising the load may be written 

f s + fi rrd m sec <p ~\ 

P r =Wr — : \ + p t W r . . (12) 

m Ur d m — fi s sec ?-> * v J 

If ? = 30 

Pr =W r m P+i-iS^O + , iWr , . (I3) 
m Ln d m — i.ispsJ c ° 

The efficiency of a triangular threaded screw, following the 
same reasoning as for square threaded screws, taking r m = r c , 

and /*!=/*, is 

W s W * d m tan « 

6 = 2itrP = *d„P = 



or e = 



SCREWS AND SCREW FASTENINGS 
W 7T d tan « 

m 

JF w d \~s + ,« - d m sec ?H ___ 

- E + /x ]y ie d m 

tan « (i — n tan « sec p) 



163 



tan « + /1 sec ^ + /* 



nearly. 



(i3 fl ) 



For the thread on a one-inch bolt in the Sellers system tan a 
= .04, and taking/* = .1, e = 11%. The efficiency of the threads 
on standard bolts is hence seen to be very low and this, as 




(a) (b) 



(d) »(e) H(f) B(g) 



(c) 



(h) 



Fig. 39. 



has been pointed out, is an advantage in fastenings as it tends 
to prevent them from unscrewing. 

56. Screw Fastenings. Screw fastenings are used for hold- 
ing two machine parts together in permanent position, or for 
adjusting one part relatively to another. There is a great variety 
of screw fastenings but all may be roughly classified as follows: 
1. Through-Bolts; 2. Studs; 3. Tap-Bolts and Cap-Screws; 
4. Machine Screws; 5. Set Screws. 

Through-Bolts. A through-bolt, or "bolt" as it is com- 
monly called, Fig. 39 (a), has a solid head on one end and a nut on 
the other. It is the best form of screw fastening and should 



1 64 MACHINE DESIGN 

always be used when the hole can be drilled completely through 
the two pieces to be held together. 

Studs. Sometimes it is not possible or desirable to drill a hole 
entirely through both pieces which are to be held together, and 
in such cases a stud-bolt, or "stud," is often used. A stud, 
Fig. 39 (b), is a circular bar having a thread cut on each end. A 
hole is tapped in the part that cannot be drilled clear through, 
and one end of the stud is screwed firmly into the hole. The 
part that can be drilled through is slipped over the stud, and 
a nut on the outer end clamps the two parts firmly together. 
Where a through-bolt cannot be used the stud is the next best 
fastening. It should be a tight fit in the tapped hole, and when 
once screwed in should not be taken out, especially if the hole is 
tapped into cast iron, as repeated removal wears out the threads. 
The length of the tapped hole should be at least one and one half 
times the diameter of the stud, in order to secure ample frictional 
resistance against turning when the nut is unscrewed. 

Tap-bolts and Cap-Screws. Tap-bolts, Fig. 39 (c), and cap- 
screws, Fig. 39 (d), have a solid head on one end and a thread on 
the other. They are used under exactly the same circumstances 
as the stud but are not as good a fastening, as they necessarily 
must be unscrewed from the tapped hole whenever they are re- 
moved. Where they have to be frequently unscrewed, and espe- 
cially if the hole is tapped into cast iron, they should be avoided. 
The only difference between tap-bolts and cap-screws is in the size 
and form of the head, the tap-bolt having a standard head (see 
next article) , and the cap-screw for the same size of bolt having a 
smaller head slightly rounded on top. Tap-bolts are much 
used in such work as securing patches on boilers, where a 
large head is desirable. Cap-screws are a little more orna- 
mental and are much used in cheaper grades of machinery. 
They are a standard article in the market and hence can be 
bought very cheaply. 

Machine Screws. Under the term " machine screws" are 
included many forms of small screws usually provided with a 
slotted head so that they may be set up with a screw-driver. The 
most usual forms of machine screws are shown in Fig. (39 e, L g 



SCREWS AND SCREW FASTENINGS 1 65 

and h). At e is shown an oval fillister head; at/ a flat fillister 
head; at g a flat countersunk head; and at h a round head. 

Machine screws are designated, for convenience, by numbers, 
the larger numbers indicating the larger diameters. Thus the 
smallest size, as given in Brown and Sharp's catalogue, is number 
000 the diameter of which is .03152. The difference in diameter 
between consecutive numbers is .01316. The diameter of a 
number o screw is .0578, so that the diameter of any number 
larger than this is given by the formula d = .0131 AT" + .0578; 
where d is the diameter in inches, and N the serial number of 
the screw. The number N is not to be confused with the number 
of threads per inch n. Machine screws larger than number 16, 
which is about %" in diameter, are not much used in machine 
work, another standard, to be discussed later, being used for 
sizes above that diameter. 

Manufacturers have, so far, been unable to agree upon stand- 
ard numbers of threads per inch, for a given diameter of machine 
screw. Thus a number 12 machine screw may have 20 or 24 
threads per inch, so that these screws are usually specified by nam- 
ing the size number first, followed by the number of threads per 
inch. Thus, an 18-20 machine screw means size 18 and 20 threads 
per inch. Because of the great confusion now existing regarding 
this point, the American Society of Mechanical Engineers ap- 
pointed a committee to establish, if possible, a system of standards 
for machine screws. This committee has reported and their 
recommendations can be found in Vol. 28 of the Transactions. 

Set Screws. Set screws are a form of screw-fastening fre- 
quently used to prevent relative rotation of two machine parts. 
Thus in Fig. 40 the hub a is prevented from revolving on the 
shaft b by the set screw c. The head of the set screw is square 
while the point may be cup-shaped as in Fig. 40 (a) , round as in 
Fig. 40 (b) , or conical as at c in Fig. 40. When the set screw is made 
in the form shown at Fig. 40 (a) , the point is hardened to enable it 
to cut into the shaft, thus increasing its holding power. If the 
screw is made of tool steel the hardening may be done by the 
ordinary process of tempering; if made of wrought iron the same 
result may be obtained by case hardening. The objection to 



i66 



MACHINE DESIGN 



the cup-shaped end is that it makes a burr on the shaft which 
sometimes greatly interferes with the removal of the hub. To 
obviate this a small conical depression is sometimes made in the 
shaft with the end of a drill and the form shown at Fig. 40 used. 
Set screws are not reliable for heavy work and should be used 
only when the load is light. 

Standard Screw Threads. Sellers or U. S. System. Screw 
fastenings larger than X mcn diameter are made according to 
some standard system in order to secure interchangeability. 
The first system of this kind was that introduced into England 




Fig. 40. 




by Sir Joseph Whitworth. The form of the Whitworth thread 
is shown in Fig. 37 (c). The thread angle is 55 and the top and 
bottom of thread are rounded off as shown. 

The recognized standard screw thread in the United States is 
the Sellers, U. S., or Franklin Institute thread. The form of 
this thread is shown in Fig. 39 (b). The thread angle is 6o°; 
the top of the thread is cut off and the bottom of the thread 
filled in as shown. This standard is not used exclusively in 
this country, however, but a full V thread, as shown in Fig. 37 (a), 
without the flattened tops and bottoms is also in common use. 
The angle of such V thread is generally 6o° in machine bolts 
and the number of threads per inch usually corresponds to those 
of the Sellers system, but there are many variations in this par- 
ticular. Where the Sellers standard is not strictly adhered to 



SCREWS AND SCREW FASTENINGS 



167 



it is advisable, therefore, to buy machine bolts of one manufac- 
turer only or so specify as to insure interchangeability. 

The Sellers screws have much greater tensile strength than 
screws with full V threads of equal angles and pitch, because the 
thread of the former is only three-fourths as deep owing to the 
flattening at the tops and bottoms. 

TABLE x 

SELLERS, U. S., OR FRANKLIN INSTITUTE STANDARD BOLTS 





Bolts 


and Threads. 


Hex 


. Nuts 


and Heads. 


Sq. Nuts and 
Heads. 


O 


u 

0) 


C8 "O 


3*3 




to 

to 




15 





H to' 

1 Co CD 


to 
to 



Ih 


u 

So 


a 

to 

co*5 
•IS 1 - 1 




rig 

§0 




J3 w 
|2<S 


to 
to 
<U 

C • 


to 
to 
V- 


m c a5 


.2 » 


P 


H 


Q 


< 


£ 


^ 


H 


H 


Inches. 


No. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


[Inches. 


1 
4 


20 


•185 


. 027 


1 
5 


37 

6T 


1 
4 


3 

16 


1 
2 


7 
TO" 


5 

16 


18 


. 240 


•045 


19 
3 2 


11 . 

16 


5 

16 


1 
4 


1 9 
32 


10 
12 


1 


16 


.294 


.068 


11 
16 


5 1 
64 


3 

8 


5 

16 


H- 


63 
6 J 


7 

T6 


14 


■344 


•093 


25 
33 


9 
TIT 


T6" 


3 

8~ 


25 
32 


*& 


h 


r 3 


.400 


. 126 


7 
8 


I 


1 


T6 


7 
8 


itt 


9 
T6 


12 


•454 


. 162 


31 

3 2 


li 


9 
16 


* 


3 1 
32 


i« 


5 

8 


1 1 


•507 


. 202 


it 1 * 


T- 7 - 


5 

8 


T6 


itV 


i| 


3 
4 


10 


. 620 


.302 


1* 


itV 


3 
4" 


1 1 

16 


I* 


T 49 
J 6¥ 


7 
8 


9 


•731 


.420 


IT 7 6 


T 21 

x 32 


7 
8 


13 
16 


itV 


" ! 32 


I 


8 


•837 


•550 


If 


I* 


I 


15 
T6 


It 


2-*4 
2 6¥ 


I* 


7 


.940 


. 692 


1+t 


' J 32 


I* 


itV 


T 13 
1 16 


2 T6 


li 


7 


1.065 


.890 


2 


2 T6 


I* 


IT% 


2 


2- 5 -3- 
2 6¥ 


I* 


6 


1 . 160 


I057 


^16 


2^1 


If 


lA 


2- 3 - 

J 16 


332 


I* 


6 


1. 284 


1.293 


2^- 
^8 


2f 


I* 


I T '6 


o 3 - 

2 8 


-,23 
3ft¥ 


1^ 


5i 


1.389 


1. 510 


2- 9 - 
r 16 


~31 

" s 32 


T 5 - 
A 8 


T 9 

IT6 


^16 


3f, 


I* 


5 


1. 491 


I. 741 


at 


2-3- 

016 


If 


T ll 

I 16 


,3 
2 4 


^57 
3e¥ 


if 
2 


5 
4* 


1 . 610 
1 . 712 


2.050 
2.300 


2 15 
z 16 

3* 


,13 
332 

38 


If 
2 


T L3 
Il6 
T l5 

Il6 


2 44 

2 16 

38 


43"2 

4*? 


2\ 


4* 


1 . 962 


3°3° 


34 


4T6 


a* 


2T6 


1 

3? 


^61 

4e¥ 


4 


4 


2 . 176 


3-7I9 


3* 


4* 


a* 


2T6 


3l 


r>31 

56¥ 


o 3 - 
z 1 


4 


2. 426 


4. 620 


4* 


,1 29 
43^ 


Z 4 


oil 
2 T6 


4i 


6 


3 


3l 


2 . 629 


5.428 


4l 


58 


3 


0I2. 
^16 


4l 


A2.1 
°32 



The area of a 1" full 6o° thread is .482 square inches, while 
the area at the bottom of a Sellers thread is .55 square inches, 
or 14 per cent greater. 

The Whitworth system of threads differs from the Sellers in 
the shape of the thread, as noted above, and the number of 
threads per inch is also different for some diameters. Thus the 



t68 machine design 

Sellers system gives 13 threads per inch for %" bolts while the 
Whit worth gives 12. The Whitworth system gives somewhat 
stronger screws as the diameter at the root of the thread is 
greater for the same size of bolt and the rounded shape at the 
root is stronger than the flat root of the Sellers thread. The 
Sellers thread is, however^ much easier to produce than the 
Whitworth. 

Table X gives the proportion of screws as fixed by the Sel- 
lers standard for bolts up to 2%". Above this size the stand- 
ard is not adhered to rigidly, as the size and pitch of the screw 
becomes rather large for convenience. Thus a 6" bolt in the 
Sellers system will have 2X threads per inch. It is common, 
therefore, to make these larger sizes, which are comparatively 
rare, with 4 threads to the inch. 

In Germany, France, and other European countries other sys- 
tems are in use, some of which are based on metric units. 

57. Pipe Threads. The Briggs system of pipe threads is 
the established standard in the United States. The numbers of 
threads per inch for the various sizes of pipe are given below: 

%" pipe, 27 threads per inch. 
%" and y%" pipe, 18 threads per inch. 
%" and }4" pipe, 14 threads per inch. 
1" to 2" pipe, 11K threads per inch. 
2y 2 " and over, 8 threads per inch. 

For form of threads and other details as to Briggs system, 
see Trans. A. S. M. E., Vol. VIII, page 29. 

58. Straining Action in Bolts due to External Load. The 

load applied to the bolts is generally one which tends to separate 
the connected members, in the direction of the axis of the bolt, and 
this action is resisted by a tensile stress in the bolts; but bolts are 
sometimes used to prevent the relative translation of two or more 
pieces, when a shearing stress is produced in the bolts. When 
the action of the load is oblique to the axis, the stress in the bolt 
may be combined tension and shearing. 

If any screw is tightened up under load there is an initial 
direct stress (tension or compression) arid usually a torsional stress 



SCREWS AND SCREW FASTENINGS 1 69 

due to friction between the threads of the screw and the nut. 
With bolts or studs screwed up hard, as in making a steam- 
tight joint, the initial tension due to screwing up may be much 
in excess of that due to the working load. This will be treated 
more fully later. 

If the load applied to the bolt produces a shearing action, the 
bolt shank should accurately fit the holes in the connected pieces, 
at least for the portions near the joint; and if P is the load per bolt, 
d the diameter of the bolt (shank), and p a the shearing stress, 

In a bolt subjected to a load which produces tension, the mini- 
mum cross section sustains the greatest stress. This smallest 
cross section, in common bolts, is through the bottoms of the 
threads. Thus if a load P be applied to an eye-bolt, as in Fig. 
41, the only stress that will be induced in the bolt will be that 
due to the external load P. If p be the tensile stress due to the 
load P, and di the diameter of the bolt at the bottom or root of 
the threads, 

P = i d * p.-. p = l± 2 . . .' . (I4) 

4 - a x 

Values of d x are given in Table X, page 167, for the various 
sizes of Sellers screws. For a given diameter and pitch of screw 
the area at the bottom of threads would be considerably less with 
full V threads. 

59. Initial Tension in Bolts due to Screwing up. If the 
bolt is used simply to hold two machine parts together, as in 
Fig. 39 (a), and there is no external load tending to separate the 
parts, the stress in the bolt will be the resultant of the tensile 
stress due to screwing up the nut, and the torsional stress due 
to the frictional resistance at the thread. 

In the Sellers system the pitch angle of the thread (a) varies 
from about 3 in a %" screw, to i°-5o' in a 3" screw; or tan « 
varies from .054 to .032 in this same range. If, therefore, in 
equation (12), r e be taken equal to | r m , a be taken at .1, and n t 
.15, it appears that P varies from .35 W with a X" screw to .32 



170 MACHINE DESIGN 

W with a 3" screw. The coefficients of friction will vary much 
more than this, so it may be assumed that for the ordinary range 
of screw fastenings 

P = .33 W approximately 

P 

or the tension in the screw W = — . 

•33 

The turning moment, Pr m , due to the wrench pull, is resisted 
by the frictional moment of the nut or collar and the frictional 
moment at the thread. This frictional moment at the thread is 
transmitted to the body of the bolt, so that the bolt itself is sub- 
jected to a twisting moment equal to Pr m minus the frictional 
moment at the nut or collar. The resultant stress, therefore, 
under these circumstances, is that due to combined twisting and 
direct stress, and it can be shown (see Art. 67) that the resultant 
stress as determined by equation (1), page 48, is from 15 to 20 
per cent greater than the direct stress alone. 

Refined calculations regarding the resultant stress in bolts due 
to screwing up are, in general, useless and misleading, especially 
in the case of fastenings less than y%" in diameter. Since a me- 
chanic using a wrench of ordinary proportions can easily rupture 
any of these small fastenings, it follows that the actual stress in 
such bolts depends entirely on the judgment of the mechanic. 

A series of experiments was made in the Sibley College Labo- 
ratory, a few years ago, to determine directly the probable load 
produced in standard bolts when making a tight joint. The sizes 
of bolts used were % n \ ffi, 1" and \%". One set of experiments 
was made with rough nuts and washers, and another set with the 
nuts and their seats on the washers faced off. A bolt was placed 
in a testing machine, so that the axial force upon it could be 
weighed after it was screwed up. Each of twelve experienced 
mechanics was asked to select his own wrench and then to screw 
up the nut as if making a steam-tight joint, and the resulting load 
on the bolt was weighed. Each man repeated the test three times 
for every size of bolt, and each had a helper on the 1" and i%" 
sizes, The sizes of wrenches used were 10" or 1 2" on the %" bolts 
up to 18" and 22" on the iX" bolts. The results were rather dis- 



SCREWS AND SCREW FASTENINGS 171 

cordant, as should be expected; the loads in the different tests 
were rather more uniform, as well as higher, with the faced nuts 
and washers. The general results indicate: (a) that the initial 
load due to screwing up for a tight joint varies about as the 
diameter of the bolt; that is, a mehanic will graduate the pull 
on the wrench in about that ratio, (b) That the load produced 
may be estimated at 16,000 lbs. per inch of diameter of bolt, or 

W = 16,000 d (15) 

in which W is the initial load in pounds due to screwing up, and 
d is the nominal (outside) diameter of the screw thread. This 
value of W is rather above the average for the tests; but it is 
considerably below the maximum, and it is probably not in ex- 
cess of the load which may reasonably be expected in making a 
tight joint. 

If the initial load due to screwing up be divided by the cross- 
sectional area of the bolt at the bottom of the threads, the initial 
intensity of the tensile stress is obtained. The above experi- 
ments indicate that this intensity of stress varies, approximately, 
inversely as the nominal diameter (d) of the bolt; and that it 
may frequently equal or exceed 

30,000 „ . ,- . 

p = ^— - — lbs. per sq. in (16) 

In addition to this tensile, stress there is, as before stated, a 
considerable twisting action on the bolt. Equation (16) would 
give a stress of 60,000 lbs. per square inch on a K-inch bolt; 
and this result is substantiated by the fact that steel bolts of this 
size were broken in the course of the experiments. It also agrees 
with common experience which forbids the use of screws as small 
as X-inch for cases requiring the nuts to be screwed up hard. 

In these experiments, the average effective lever arm of the 
wrench was not far from 15 times the diameter, or 30 times the 
radius, of the screw; hence, if it be assumed as in the previous 
paragraph that the turning force acting at the radius of the screw 
is P = .3$W the force applied at the wrench is, in pounds, about 

P ,-itW .33 X 16,000 d 
P. = — = -^ — = — ■ = 176 (J. 

3° 3° 3° 



172 MACHINE DESIGN 

The above discussion indicates that the factor of safety should 
be increased as the size of the screw decreases, and of course this 
factor should be varied with the conditions of the case, as in 
some applications the nuts are much more apt to be screwed up 
hard than in others. 

A set of experiments was made by Mr. James McBride (Trans. 
A. S. M. E., Vol. XII, page 781), which show that the factor of 
safety, as bolts are frequently used, is very low, even with a very 
moderate external load. One case cited by Mr. McBride indi- 
cates that the stress due to screwing up a 3 yi -inch bolt was 
nearly one-half the ultimate strength, or probably very near the 
elastic limit. His direct determinations of the efficiency of a 
standard 2-inch screw bolt shows an average of only 10.19 per 
cent. It is probably this low efficiency which saves many screws 
from being broken, as the frictional loss reduces the tension pro- 
duced in the bolt by screwing up. The excessive friction makes 
the screw bolt a useful fastening, as it reduces the tendency to 
"overhaul" or unscrew. 

60. Resultant Stress on Bolts due to Combined Initial Tension 
and External Load. It was shown, in Art. 59, that bolts 
may be subjected to a high tensile stress by screwing up the 
nuts. The question often arises as to the effect of the combined 
action of this initial tension and the external, or useful, load. 
It is stated by some that the resultant load on the bolt is simply 
the sum of the initial and the external loads. Others contend 
that the application of the external load does not change the 
stress in the bolt, unless this external load exceeds the initial load 
due to screwing up; that is, that the resultant load is equal to 
the initial load alone, or to the external load alone, whichever is 
the greater. 

Neither of these views is entirely correct for conditions at- 
tained in practice. They represent the extreme limiting cases 
and every actual case lies between them. 

If the bolt itself could be absolutely rigid while the members 
forced together in screwing it up yielded under pressure, the total 
load on the bolt would be equal to the sum of the initial load and 
the external load. If, however, the members pressed together 



SCREWS AND SCREW FASTENINGS 



J 73 



were absolutely rigid, only the bolt yielding, the total (resultant) 
load on the bolt would be the initial load alone, or the external 
load alone, whichever is the greater. 

The first of the above conditions is approached by the arrange- 
ment shown in Fig. 42. Screwing up the nut compresses the 
spring interposed between A and B. Assume that an axial force 
of 2,000 pounds will compress this spring 1 inch; then if the nut 
is screwed up till the spring is 2 inches shorter than its free length, 
the load on the bolt, due to screwing up, must equal the reaction 





Fig. 47. 



Mr) 



Fig. 46. 





II 




\ 


'•L 






/ 


Fig. 48. 


, 




Fig. 42 



Fig. 43. 



Fig. 44, 

J I 

Fig. 45- 




of the spring, or 4,000 lbs. Assume, also, that the extension of 
the bolt under this screwing-up action, or under the initial load 
of 4,000 lbs., is .02 inch. Now, if an external axial load of say 
2,000 lbs. be applied to the eye at the bottom of B, this added load 
would tend further to increase the length of the bolt by about .01 
inch; but this further extension of the bolt would reduce the 
compression on the spring by a corresponding amount and thus 
slightly diminish the spring reaction. With such great differ- 
ence between the rigidity of the bolt and of the connected mem- 
bers, the load on the bolt becomes practically the sum of the 



174 MACHINE DESIGN 

initial and the external loads, but the resultant load is necessarily 
somewhat less than this sum in any possible case. 

The arrangement shown in Fig. 43 is one which approaches 
the other limiting case mentioned above. Suppose the bolt to be 
a spring which is subjected to an axial load of 4,000 lbs. in screw- 
ing the nut up two inches, and that the corresponding yielding of 
the member B is .02 inch. The initial load on the bolt (which is 
the spring in this case) is 4,000 lbs., and the pressure between 
the contact surfaces of A and B is equal to it. If an external 
axial load be now applied to the eye in B, the pressure between 
the contact surfaces is reduced by an amount nearly equal to this 
external load. But, unless the external load exceeds the initial 
load, the bolt will not elongate enough to separatej:hese contact 
surfaces and entirely remove the pressure between them, because 
the load on the bolt (spring) cannot change without changing the 
length of the bolt, and with the above data the bolt would have to 
stretch an additional .02 inch (equal to the initial yielding of the 
connected members) before the contact surfaces would be entirely 
relieved of pressure. It therefore appears that the addition of an 
external load in this case does not materially affect the resultant 
tension on the bolt as long as this external load does not exceed 
the initial load. If the external load is greater than the initial 
load (say 6,000 lbs.), the elongation of the bolt increases (to 3 
inches) ; the resultant load on the bolt will be simply the external 
load alone, because the latter is sufficient entirely to relieve the 
pressure produced between the contact surfaces in screwing up. 

In all ordinary practical cases the difference in rigidity between 
the bolt and the connected members is much less than in the ex- 
treme conditions considered above. The resultant load on a bolt 
may be anything between the sum of the initial and the external 
loads as a maximum, and the greater of these two loads alone as a 
minimum. This resultant load approaches the maximum limit 
when the bolts are rigid relative to the connected members as in 
Fig. 44; and this resultant approaches the minimum limit when 
the bolts are relatively yielding, as in Fig. 45. In any particular 
case the designer can tell which limit is the more nearly ap- 
proached, and he should be governed accordingly. 



SCREWS AND SCREW FASTENINGS 1 75 

The Locomotive (Nov., 1897) contains an excellent article 
on the resultant load on bolts, and a relation is derived from 
which the following method of treatment has been developed: 
The application of this method depends simply upon the ratio of 
the yield of the connected members to the yield of the bolts. It 
will usually not be difficult to assign a sufficiently close value to 
this ratio even when the actual magnitudes of yielding are un- 
known; in fact, only a rough approximation to the value of this 

y 

ratio is necessary. Let this ratio be called y and let = x; 

call the initial load on the bolt due to screwing up W \; the exter- 
nal (useful) load W 2 ; and the total (resultant) load W. Then it 
can be shown that 

W = W t + x W 2 . 

If the yield ratio (y) is known, the value of x is at once found by 
the above relation of x and y. If the yield of the connected mem- 
bers is between 1 and 5 times that of the bolt, the resultant load 
is equal to the initial load added to from 0.5 to 0.8, the external 
load. If a tight joint is made with short rigid bolts or studs, con- 
necting flanges which are separated by an elastic packing, or with 
a metal contact at some distance from the center line of the bolts, 
as indicated in Fig. 44, the applied load is an important consider- 
ation since the value of y is relatively great. In some other cases 
the external load may be a minor consideration as affecting the 
strength of the bolt. 

When the conditions are such that the nut is not apt to be 
screwed up hard, that is when the initial load may be safely 
neglected, design for the external load alone. 

The following suggestions may serve as a guide in practical 
problems involving the resultant load on bolts when the initial 
load due to screwing up is apt to be considerable. 

(a) If a bolt is manifestly very much more yielding than the 
connected members, design the bolt simply for the initial load or 
for the external load, whichever is the greater. 

(b) If the probable yield of the bolt is from one-half to once 
that of the connected members, consider the resultant load 



176 MACHINE DESIGN 

as the initial load plus from one-fourth to one-half the external 
load. 

(c) If the yield of the connected members is probably four or 
five times that of the bolts, take the resultant load as the initial 
load plus about three-fourths the external load. 

(d) In case of extreme relative yielding of the connected mem- 
bers, the resultant load may be assumed at nearly the sum of the 
initial and external loads. 

61. Allowable Stress in Screw Fastenings. From the fore- 
going it is seen that small screw fastenings are very liable to be 
heavily overstrained by the initial load due to screwing up the 
nut. While the body of the bolt is well designed to resist heavy 
loads a source of weakness is found in the threaded portion. 
The reduced area, due to cutting the thread, localizes the greatest 
stress, and cracks are very liable to start from the roots of the 
threads, especially where the thread is of the full V form. 

For these reasons the ordinary apparent fibre stresses allowed 
in most machine members cannot be permitted in screw fasten- 
ings. For ordinary purposes, where overstraining is not likely 
to occur, or for large bolts, 8,000 to 10,000 lbs. per square inch 
may be allowed, for steel. For such work as steam and hydraulic 
joints, where the initial stress may be large, from 6,000 to 8,000 
lbs. per square inch should be allowed, depending on the condi- 
tions and quality of material employed, and if shocks are liable 
to occur, stresses as low as 3,000 to 4,000 are often preferable. 

Example 1 : The cylinder of the steam engine is 1 2 inches in 
diameter, and the cylinder head is held in place by 10 steel 
through bolts. The maximum steam pressure is 100 pounds per 
square inch. If the contact surfaces of the head and cylinder 
are ground together so that no packing is necessary, what must 
be the diameter of the bolts so that the maximum stress in the bolt 
necessary to insure a steam-tight joint will not exceed 7,000 lbs. 
per square inch ? 

In this case it is evident that the bolts are much more yielding 
than the parts which they hold together and the conditions are 
those of case a in the previous paragraph. It is also clear that 
the initial load on the bolt must be greater than the external 



SCREWS AND SCREW FASTENINGS 177 

load due to the steam pressure in order to insure a steam-tight 
joint. If this initial load be taken at twice the external load a 
fair margin of safety is secured. If W t be the initial load and 
W 2 the external load per bolt then 

7T I2 2 

W 2 = X 100 = 1,130 lbs. 

2 4 X 10 6 

whence W l — 2 W 2 = 2,260 lbs. 

Whence if d t be the diameter at the root of the thread 

*d? 

X 7,000 = 2,260. 

4 

Therefore d x = .64 inch (at root of thread) 
which corresponds closely to a W screw. It is to be noted that 
while a total stress of 7,000 lbs. per square inch of section is 
sufficient to insure a tight joint, a much greater stress may be in- 
duced by the workman in screwing up the nut, if he is careless 
or inexperienced. 

Example 2 : If in the above example steel studs are used and 
rubber packing y&" thick be placed between the contact surfaces, 
what must be the diameter of the studs ? 

Here the parts held together are more elastic than the studs 
and the conditions may be taken as corresponding to those of 
case c. As before, the initial load W x may be taken at twice the 
external load. 

" X 12 2 

Then W 2 = X 100 = 1,130 

2 4 X 10 

and W x = 2 W 2 = 2,260. 

From (c) paragraph 60 the total load 

W = W x + X W 2 

= 2,260 + (K X 1,130) = 3,107 lbs. 

Whence X 7,000 = 3,107 

4 

and d x = .76 inch (at root of thread) 
which corresponds closely to a yi inch screw. 
12 



178 MACHINE DESIGN 

The maximum stress which the workman may, perhaps, in- 
duce in the stud by screwing up the nut is 

30,000 30,000 
p = — -z — = — — — = 34>ooo lbs. approximately, 
a y% 

which will be increased a little by the external load. This is 
close to the elastic limit; but it may be noted that even should 
the elastic limit be slightly exceeded the efficiency of the fast- 
ening is not impaired, since here permanency of form is not so 
essential as in machine parts which transmit motion. 

62. Resilience of Bolts with Impulsive Load. 

In bridge work and other cases requiring long bolts, it is very 
common to make the cross-section through the body of the bolt 
about equal to the section at the bottom of the threads. This 
may be done by upsetting the ends where the thread is to be cut, 
or by welding on ends made from stock somewhat larger than 
that used for the main length of the bolt. 

The most apparent result of this practice is to economize mate- 
rial without sacrifice of strength (as the shank still has an area of 
cross-section equal to the threaded portion), and if the weld 
(when the ends are welded) is perfect, the strength of the bolt 
is not reduced. It seems probable that this reason is responsible 
for the original adoption of this practice, since it has been most 
generally used in long tie rods. However, in case of bolts liable 
to shock, there is an even more important reason for such con- 
struction ; since it can be shown that the reduced section not only 
maintains the full strength under static load, but it very greatly 
increases the capacity of the bolt to resist shock. This last fact 
has not been very generally recognized, as appears from the com- 
mon application of such reduced shank bolts only to structures, 
rather than to machines. 

It has been seen that the resistance of a tension member under 
a static load is determined solely by its weakest section; while, 
in a member subjected to shock, impact, or impulsive load, the 
resistance depends upon the total extent of distortion of the 
member due to a given intensity of stress. 

As shown in Art. 24, the maximum stress with impulsive load is 



SCREWS AND SCREW FASTENINGS 1 79 

W (h + d) 
P " k d A 

For a stress within the elastic limit 

2W ( h + <5 \ 2W f h \ 

This shows clearly that for a given load, W, applied suddenly 
or with impact, the stress produced in a member of sectional area, 
A, is greater as £ becomes less relative to h. Hence, if d is in- 
creased, the stress produced becomes less for a given impulsive 
action; or the resistance to such action is greater for a given 
value of the maximum stress. 

If an ordinary bolt is subjected to shock in a direction to pro- 
duce tension, the stress will be a maximum at the sections through 
the bottom of the threads; the bolt will elongate, but the elonga- 
tion will be confined largely to the very short reduced (threaded) 
sections, hence the stress will be much less in the larger portion 
of the bolt. In a Sellers bolt of one inch diameter the area A of 
the shank is .78 sq. inches, while the area A' at the bottom of 
threads is only .55 sq. inches. Therefore a stress on A' of 30,000 

lbs. per sq. in. = — — — = 21,000 on the full sections. Sup- 

.78 

pose the elongation per inch of length at a stress of 30,000 (taken 
as the elastic limit) is two""* Each inch of section A' will elongate 
Trrro", while each inch of full section A ( = .78 sq. in.) will have 
a stress of only 21,000 lbs., with a corresponding elongation of 
It X ToVo = - 000 7"- Assume the thread to be 1" long, and the 
remainder of the bolt to be 5" long. It will appear that the mean 
stress on the threaded portion (1") is about the mean of 30,000 
and 21,000, or say 25,500 lbs. per square inch; as the mean sec- 
tion is an average of .55 and .78 square inches. Hence the 
elongation for this threaded 1 inch, when the stress on 4' = 30,- 
000, is .00085", while the other 5" (of area A) will elongate under 
this load 5 X .0007 = .0035". The total elongation will then te 
3 = .00085 + .0035 = .00435 inches. 



l8o MACHINE DESIGN 

A' p d 



iih = T y, w = 



h +d 



,tC X 30,000 .0043S n , ' mm 

— — x — ^~ = 8250 X .0416 = 344 lbs. 

2 .10435 

Now, suppose the 5" shank of this bolt were reduced in section 
to an area A' = .55. Then the elongation of this portion under 
the above load would be 5 X .001 = .005", instead of .0035" and 
the total elongation would be d = .00085 + .005 = .00585. 

•55 X 30,000 .00585 n „ 

•'• W = • , X —^r = 8250 X 0.553 = 457 lbs. 

2 -10585 

This latter load is 33 per cent greater than the preceding. 

The preceding example shows that the elastic resilience of the 
bolt was increased 33 per cent by reducing the body of the bolt 
to A' '. Of course the gain would be still greater with a longer 
bolt. It may be well to remember that the "long specimen " is 
more apt to contain a weak section than is a short specimen ; but, 
on the other hand, the sharp notching of the threads is quite 
liable to start a fracture at their roots. 

If the bolt is strained beyond the elastic limit, the portion thus 
strained yields at a much greater rate, relative to the stress, than 
that given above. With a load which would produce a stress of 
30,000 lbs. per sq. in. in the larger portion (area A), the stress in 

30,000 X .785 
the reduced portion (area A') will be = 43,000 lbs. 

per sq. inch. Hence, the effect of a long section in resisting 
shock without rupture is much greater even than that shown for 
elastic deformation only. 

The section of the shank of the bolt may be reduced as in Fig. 
46, by turning down the body of the bolt to about the diameter 
at the bottoms of the threads. The collars a and a' may be left 
to form a fit in the hole. This form is easy to make, but does 
not fit the hole throughout its length, and it is weak in torsion. 

Fig. 47 is somewhat more expensive, but fits the hole better, 
and is somewhat stronger in torsion. Fig. 48 is the form which 
gives the best fit, and is also the strongest in torsion. If very 
long it is difficult to make; otherwise it is perhaps the best. 



SCREWS AND SCREW FASTENINGS l8l 

These high resilience bolts only increase the resistance to im- 
pulsive load, not to dead load. They are good forms to use in 
such cases as the so-called " marine type" of connecting rod, 
where the bolts are subjected to considerable shock. 

For cylinder head bolts, and other cases where a tight joint is 
the main consideration, this form of bolt may be entirely unsuited. 

Professor Sweet prepared, for tests, some bolts such as are used 
in the connecting rod of the Straight Line Engine; of these, half 
were solid (ordinary form) bolts, and the other half were of the 
form shown in Fig. 48. 

Tests of a pair of these bolts, one of each kind, showed an elon- 
gation at rupture of .25* for the solid bolt, which broke in the 
thread; while the drilled bolt elongated 2.25", or 9 times as 
much, and it broke through the shank, the net section of which 
was a trifle less than that at the bottom of the threads. Drop 
tests showed similar results. These tests indicate the superior 
ultimate resilience of the reduced shank bolts. 

It was shown in Art. 24, page 77, that where a machine 
member must absorb considerable shock, a rather weak yielding 
material might be safer than one which is stronger and stiffer, 
because of the greater elastic resilience of the weaker and more 
ductile material. This principle is of importance in designing 
fastenings which are subjected to shock where they must neces- 
sarily work under high stress. 

63. Location of Fastenings. i\s previously stated, screw 
fastenings are generally intended to be tension members only, 
and from the foregoing discussion it appears that even when used 
in this manner alone they are subjected to very high stresses. 
The conditions under which a fastening is to be used should 
therefore be carefully considered in order that all forces acting 
upon it may be provided for. Further, the location of the 
fastening may or may not be advantageous, thus greatly affect- 
ing its required size. Thus in Fig. 49, if the bolts alone are de- 
pended upon to resist the downward force P, they must be care- 
fully fitted, to insure that each bolt receives its full share of this 
shearing load. Through bolts only can be used in such a case 
as studs or tap bolts cannot be accurately fitted. If the down- 



l82 



MACHINE DESIGN 



ward force is resisted by a projecting ledge, as at A, which is 
the preferable way, the bolts need not fit the holes closely and 
either studs or tap bolts can be used. The bracket now tends 
to rotate around A and the moment of the load P I must equal 
the sum of the moments of the bolts round the same point. It 
is evident that the lower bolt must be considerably larger than 
the upper bolt, to be equally effective. In small work it is con- 
venient to make all bolts the same size, the sum of their resist- 
ing moments being made equal to P I. In large work the bolt 
at C is often made large enough to exert a moment equal to P I, 
and the bolts near A serve only to insure correct location. The 
upper bolt should, in any case, be located as far away from A as 
possible. 




Fig. 49. 



In many machine parts, such as flywheel rims and brake bands, 
it often occurs that the bolts cannot be placed directly in line 
with the applied force but must be at a distance / (Fig. 50) from 
its line of action. The bolts in such cases may be subjected to 
both flexure and direct stress. Thus in Fig. 50, if the bolts fit 
the holes in the lugs tightly such a combination of stresses will 
be induced. In such parts as brake bands the connecting bolts 
are often used as a means of adjustment against wear, as shown 
in Fig. 51. If in such a case the lugs be weak and yielding, the 
threaded portion of the bolt will be subjected to both flexure and 
direct stress. The threaded portion of the bolt is particularly 
weak against flexural stress because cracks are easily started at 
the root of the threads, and where screws are used in this manner 
they should be designed with a large factor of safety. 



SCREWS AND SCREW FASTENINGS 183 

64. Screws for the Transmission of Power. It has been 
pointed out, in Art. 53, that the square thread is most used for 
transmitting power because of its higher efficiency, and that 
when wear must be compensated for the half V thread is most 
serviceable (see Figs. 37 d and e ). Where the thread angle of the 
half V thread is small, as in the Acme thread, the general equa- 
tions which have been deduced for the square thread may be 
used without great error. 

Equation (6), Art. 54, expresses the relation which exists be- 
tween the turning moment which must be applied to the screw 5 
and the moments due to the load, friction at the thread and at the 
thrust collar. An examination of this equation shows that for a 
given applied force P, the load W which can be overcome de- 
creases with an increase in the value of the pitch s, and in- 
creases with a decrease in the value of s, since s is added to the 
numerator of the fraction and subtracted from the denominator. 
This can be seen in another way by considering the energy sup- 
plied and the work performed. If the force P be applied 
through a complete revolution, or a distance of - d m , the load 
will be raised a distance equal to the pitch s. Evidently, if 5 is 
decreased, a greater load can be raised by a given force P ; since 
(neglecting friction), the force applied, multiplied by the space 
through which it moves, must be equal to the load multiplied by 
the space through which it is raised. In other words, the me- 
chanical advantage of the screw can be varied by reducing the 
pitch angle; and it is evident that by reducing the pitch angle a 
small force applied at a long radius may be made to raise a great 
load. 

In order that the thread on the screw and nut may be equally 
strong, with similar materials, the thread and space on the screw 
are made equal to each other and therefore equal to half the 
pitch. As the pitch is increased the axial width of both thread 
and space are necessarily increased, and if it is desired to keep 
the section of the thread square in form, this soon results in a 
very heavy thread when compared to the cylinder on which it is 
formed. If the depth of the space is reduced, to avoid reducing 
the diameter of the cylinder, the bearing surface of the screw 



184 MACHINE DESIGN 

and nut is reduced, which is not desirable. It is customary in 
such cases to divide the axial width of the thread and space into 
several equal parts, arranged alternately round the axis of the 
screw, thus forming several parallel threads and spaces. The 
depth of the space can, by this means, be greatly reduced and 
ample wearing surface be provided. Such screws are called 
multiple threaded screws and may have two, three, or more 
parallel threads. The theory of such screws is evidently iden- 
tical with that of the single threaded screw. 

65. Friction and Efficiency of Screws for Power Trans- 
mission. Equations (9) and (10), while expressing the general re- 
lations which exist between efficiency and the pitch angle, do not 
show clearly the effect upon the efficiency due to varying this 
angle. In Fig. 52 these equations have been plotted for various 
constant values of n, and these curves show graphically the effect 
of varying the pitch angle. An examination of this figure shows 
that for the value of n chosen the efficiency increases rapidly, as 
the angle increases, up to 15 or 20 , attaining a maximum be- 
tween 40 and 50 , and then decreasing with an increase in the 
angle, becoming zero again near 90 . 

It is to be noted that between 20 and 6o° the efficiency does 
not vary materially with change of angle, and that when the 
efficiency of the screw alone is considered, steep pitched threads, 
as from 30 to 50 pitch angle, give maximum efficiency and 
hence a more durable thread. It is seldom feasible to use such 
pitches in practice, for reasons that will be presently discussed. 
The curves in Fig. 52 will be found useful in making trial 
assumptions for the efficiency of screws. 

Screws for transmitting power are usually difficult to lubricate 
freely, hence, in general, their rubbing surfaces are imperfectly 
lubricated (see Art. 28). The coefficient of friction for screws 
working under pressures ranging from 3,000 to 10,000 lbs. per 
square inch, and at low velocities, has been experimentally de- 
termined * by Professor Albert Kingsbury. From his experi- 
ments it appears that, for these conditions, the value of y. may be 

* Transactions of American Society of Mechanical Engineers, Vol. 17. 



# Efficiency 

•j*. » cj 

o o o 





H 




=r 


Tj 


n 




p 




p. 




t 


t° 


3 



























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































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i 


\ 




















































































\ 








\ ' 










































1 - Screw alone /i = .05 

2 — Screw and Collar fi — .05 

3 - Screw alone //. = .1 

4 - Screw and Collar fi= A 

5 - Screw and Collar # = .15 


























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l86 MACHINE DESIGN 

taken at .15. For pressures lower than 3,000 lbs. per square 
inch, and velocities above 50 ft. per minute, the value of fi may 
be assumed at .1, if fair lubrication is maintained. 

The bearing pressure per unit area of thread surface that 
may be carried on a screw thread, will vary greatly with the 
conditions of service. If the velocity is low, and wear not an 
important factor, as in the case of jack screws, very heavy pres- 
sures may be carried; but where accuracy of form is important, 
and where the velocity exceeds 50 ft. per minute, the pressure 
per unit area should not exceed 200 lbs., and for such service as 
lead screws, where maintenance of form is essential, it should be 
as low as possible. 

If J^ = load carried. 

p = intensity of pressure per unit of projected area of thread. 
n = number of threads per inch. 
I = length of nut in inches. 
d = outside diameter of thread. 
d x = inside diameter of thread. 

Then W = pnl-[d 2 — d*] . . . . (17) 
4 

The load per unit of projected area is the same as the load per 
unit of true area, since the projected area is equal to the true 
area multiplied by cos «, and the axial or projected pressure is 
equal to the normal or true pressure multiplied by the same 
function. 

66. Stresses in Transmission Screws. It has been shown 
in Art. 59 that the resisting moment at the thread is, from 

equation (6), equal to W r \ — — — - I or to the total turning 

L -7rd m — Sfi-i 

moment applied, minus the frictional moment at the collar. 
This moment induces a torsional stress* in the screw. The 
direct action of the load is to induce a tensile or compressive 

* This statement applies strictly to the most usual case only, where the thrust 
collar is located at that end of the screw to which the power is applied. If the 
collar is not located at the end to which the power is applied, the total torque, 
Pfm, of equation (6) is transmitted through the body of the screw. 



SCREWS AND SCREW FASTENINGS 1 87 

W 
stress in the screw equal to — (where A is the area at the 

root of the thread) if the screw is short. If the screw is over 
six times as long as its least diameter d lf the compressive stress, 
if the screw is in compression, will be that due to W, considering 
the screw as a long column. Equations (1) and (2) of Chapter 
III, page 48, and their discussion in Art. 18, are therefore, ap- 
plicable to the design of such screws. 

67. Design of Screws for Power Transmission. An in- 
spection of Fig. 52 shows that screws of small pitch have very 
low efficiency, and it would seem desirable for that reason to 
keep the pitch as great as possible. On the other hand, it was 
pointed out in Art. 64 that the mechanical advantage of a screw 
increases as the pitch decreases. It was also shown in Art. 54 
that a self-sustaining screw could not have an efficiency of over 
50 per cent, and for perfect safety against overhauling it should 
be much less than this value. The best pitch, for a given set of 
conditions, may therefore be a compromise between these con- 
flicting requirements. Thus if the turning moment which can be 
applied to a screw is limited (as is often the case with hand 
power) a low pitch must be selected in order to attain mechanical 
advantage. In such a case it is obvious that care should be used 
in selecting the materials of the screw and nut, so as to obtain as 
low a coefficient of friction as possible. Thus a bronze nut will 
run well on a steel screw with imperfect lubrication. (See Art. 
28.) ' Again in such cases as the screws in certain machine 
tools, as plate planers, a more efficient pitch may be taken. If 
there is no tendency for the screw to overhaul, and the necessary 
moment can be applied, the pitch of maximum efficiency can be 
selected.* 

Example. The force required to open or close a certain sub- 
merged sliding water gate is estimated at 6,000 lbs. It is re- 
quired to design a single-threaded steel screw such that one man 

* In certain forms of saw-mill carriages these conditions exist, and the screws 
for setting the log over to the saw may be and are made with a very efficient 
pitch. 



1 88 MACHINE DESIGN 

exerting a pull of 60 lbs. at the periphery of a hand wheel 40 
inches in diameter, attached directly to the screw, can operate 
the gate. The greatest unsupported length of the screw is 
found to be 4 ft. Let the coefficient of friction = .15, the crush- 
ing strength of the material = 30,000 lbs. per square inch, the 
maximum working tensile or compressive stress = 10,000 lbs. per 
square inch, and the coefficient of elasticity = 30,000,000. 

Since the screw is in compression in closing the gate it must 
be designed as a long column square at both ends (case 4), and 
if so designed it will have surplus strength when in tension. 
The effect of the thread in stiffening the screw is small and will 
be neglected. The total maximum unit stress in the screw is to 
be 10,000 lbs. and it is evident that the maximum compressive 
stress p will form the larger part of the total stress; p may there- 
fore be taken at 9,000 lbs. per square inch, and the mean com- 
pressive stress p' assumed at 6,000 lbs. per square inch : whence 
the trial area of the screw at the root of the thread 

P 6,000 

= — ; = = i sq. inch, or a diameter of iy& inches. Check- 

p' 6,000 

ing this assumption by formula N page 94 

P p 9,000 

p = ^" I{ A ( l )^ n 3°,°°° py =5 '3° olbs - 

mn 2 E\pJ 4X^X30,000,000 V. 28/ 

which checks closely enough with the mean stress assumed. 

Assume the efficiency of the screw and collar at 1 5 per cent. 
The energy which the operator can supply in one complete revo- 
lution of the wheel = ~X 40 X 60 = 7,600 in. lbs. Hence the 
energy delivered at the nut will be 7,600 X .15 = 1,140 inch lbs. 
But during one revolution of the screw the gate must move a dis- 
tance equal to the pitch s, against a force of 6,000 lbs. 

Hence 5 X 6,000 = 1,140 

.'. s = -r = .10" or say \" = .2" so that the thread may 

6,000 

be easily cut in a lathe. 

Since the thread is square the outer diameter of the screw will 



SCREWS AND SCREW FASTENINGS 189 

be d t + s = \y&" + .2 = 1.325, or in order to use a standard tap 
the outer diameter may be taken as iK". The corrected diame- 
ter of the screw at the root of the thread will be 1.5— .2 = 1.3" 
and the corrected mean diameter will be 1.5 — .1 = 1.4". For 

.2 

these values tan « = = .046 which corresponds to 

- X 1.4 

a = 2° — 40'. From curve 5 (Fig. 52) the efficiency of the screw 
is about 13 per cent, and the original assumption of efficiency 
is sufficiently close. 

The twisting moment applied to the screw = 60X20 =1,200 
in. lbs. The frictional moment at the collar is approximately 
/i W r m = .15 X 6,000 X .7 = 630 in. lbs. Hence the torsional 
moment at the nut* 

T = 60 X 20 — 630 = 570 in. lbs. 

The torsional stress due to this moment is by equation E, 
page 91, 

16 T 16 X 570 

A, = 77 = 7 ^ = i j3 2 ° lbs. 

^ *^ 3 *x(i.3) 
.*. by equation (I), page (48), the maximum direct stress, 

P m = X IP + Vp 2 + 4 Ps 2 ] = y [9>°°o + Vo,ooo 2 + (4Xi,320 2 )] 
= 9,200 lbs., which is less than the assigned limit and the 
design is therefore correct. 

The increase in the maximum stress due to the torsional mo- 
ment is here only about 6 per cent. Where the screw is short, so 
that a much greater mean direct stress can be carried, or where 
the screw is only in tension and hence admits of a high tensile 
stress, this increase may be from 15 to 20 per cent. If the screw 
is made of cast iron it should also be checked for shearing by 
equation (2) of Art. 16. 

* It is assumed that the collar is at the upper end where the power is applied. 



CHAPTER VIII 
KEYS, COTTERS, AND FORCE FITS 

68. Forms of Keys. Keys are wedge-shaped pieces, generally 
made of steel, which are used primarily to prevent relative rota- 
tion between shafts and the pulleys, gears, etc., which they carry. 
On account of the frictional resistance which they induce between 
the surface of the shaft and the member which is keyed to it, they 
also often prevent relative sliding of the parts. Keys are most 
usually rectangular in cross-section; but occasionally they are 
made of circular form. A saddle key is shown in Fig. 53. 
This form of key does not require the shaft to be cut; but its 
holding power is so small that it is used only for light work. For 
small loads, or as a safeguard when the hub is shrunk on, a round 
pin as shown in Fig. 54 (a) is often used. Figs. 54 (b) and 54 (c) 
show two other methods of applying round taper pins as a sub- 
stitute for keys. A flat key is shown in Fig. 55. This form 
requires a small portion of the shaft to be cut away, and its hold- 
ing power is much greater than that of the saddle key. The sunk 
key, Fig. 56, is the most secure form of key fastening, and is more 
used than any other. It is so called because it is sunk into a 
keyway or groove cut in the shaft. It thus requires more metal 
to be cut away from the shaft than the flat key, and this must be 
taken into account in designing shafting since the metal is removed 
from the outer fibre where it is most serviceable for resisting 
applied loads. The keyway cut in a shaft for a sunk key is made 
parallel to the axis of the shaft; but the keyway in the hub of the 
pulley or gear which is to be made fast is cut tapering as shown 
in Fig. 56 (b). The sides of the key are parallel, as shown, and 
should fit well in both shaft and hub. When the key is driven 

190 



KEYS, COTTERS, AND FORCE FITS 



IQ1 



in, the shaft and hub are drawn tightly together on the side of 
the shaft opposite to the key, and the frictional resistance thus 
set up helps to prevent relative sliding of the parts lengthwise 
of the shaft. If the bore of the hub is tapering, or if the key fits 




Fig. 53. Fig. 54 (a). Fig. 54 (b). Fig. 54 (c). Fig. 55. 

more tightly at one end than at the other, the part keyed on may 
be thrown out of alignment so that its plane is not perpendicular 
to the axis of the shaft. Where great accuracy is required, as in 
flanged couplings on shafting, owing to this tendency, the faces 
of the flange or part secured to the shaft should be faced in place 
after the key is driven. If the part keyed on does not have to 
be removed often, the hub may be made a tight or press fit on 
the shaft, thereby preventing largely the tilting action of the key 
should such occur. In the Woodruff system (Fig. 57), the key 




Fig. 57. 



is a circular segment and the keyway may be cut with a milling 
cutter. This allows the key to adjust itself to the taper of the 
keyway in the hub, hence it will not throw the keyed part out 
of perpendicular alignment. With this system, the hub must 



I92 MACHINE DESIGN 

be forced on over the key. These keys are used largely in ma- 
chine tools. 

In general, the part to be secured on the shaft is placed in 
position and the key driven in. This makes it necessary to ex- 
tend the keyway along the shaft at least the length of the key, 
(except when the hub is at the end of the shaft) unless 
the diameter of the shaft is enlarged under the hub, suffi- 
ciently to allow the keyway to be cut without cutting into the 
shaft proper. Where it is desirable to withdraw the key occasion- 
ally, it is often provided with a head, as shown in Fig. 56 (b), in 
which case it is called a draw key.* Sometimes, however, it is 
not desirable to extend the keyway beyond the hub, in which 
case the keyway in the shaft is made the same length as the key, 
and the hub is driven over the key into its correct position. 
Much more force is necessary to drive the hub into place in this 
manner than to drive the key, on account of the friction between 
the shaft and the hub. When the hub is a sliding or an easy fit 
on the shaft, and only one key is used, there is a tendency to 
throw the hub eccentric to the shaft. Under these circumstances 
there is a tendency for the hub to rock and work loose on the 
•shaft, especially if the direction of motion be reversed. In such 
cases two keys set 90 apart make a much more secure fastening 
as this gives three lines of contact and prevents rocking. If one 
of these keys is a saddle key, as shown in Fig. 53, the fitting is 
greatly facilitated and the fastening is almost as secure as with 
two sunk keys. 

69. Stresses in Sunk Keys. Since keys are designed to prevent 
relative rotation, it is evident that every key must transmit a 
certain torsional moment or torque. This torsional moment 
may be equal to the total torque transmitted by the shaft, or the 
key may be required to transmit only a part of it. This would 
indicate that keys of different sizes should be used with any given 
diameter of shaft, depending on the load which the key must 
transmit. For practical reasons, however, such as standardiza- 



* Where a draw key cannot be used the point of the key is sometimes case- 
hardened so that it will not upset so readily in being driven out. 



KEYS, COTTERS, AND FORCE FITS 193 

tion and interchangeability, it is desirable that the dimensions 
of the shaft and key should bear a fixed relation to each other. 
All practical systems of keys, therefore, give a fixed size of key 
for each diameter of shaft, the dimensions of the key, presuma- 
bly, being such that its strength is equal to the torsional strength 
of the shaft. Shafts are usually designed for torsional stiffness 
rather than torsional strength, which results in a shaft consider- 
ably larger than necessary as far as strength is concerned. If, 
under these circumstances, the key is designed as indicated above 
it will also have excess strength. Where the shaft is short and 
is designed for strength alone, the key should be more carefully 
considered. 

Keys resisting a torsional moment are subjected to simple 
crushing, or to crushing and shearing, depending on the man- 
ner of their application and manner of fitting. The ordinary 
sunk key (Fig. 56 a), is subjected to a force, F ly due to the 
pressure from the shaft, and to a resisting force, F 2 , due to 
the reaction from the hub which it secures. The effect of 
these two forces is to set up a shearing stress along the middle 
section of the key at the outer surface of the shaft. They also form 
a couple which tends to rotate the key in the keyway. This tend- 
ency to rotate should be, for best results, resisted by the pressure 
of the hub and shaft against the top and bottom of the key. If 
the key is not a tight fit on the top and bottom these resisting 
pressures, F 3 and F 4 , will be concentrated near the corners. This 
concentrated pressure may be sufficient to crush the key at these 
points, and allow it to roll in the keyway, deforming both the 
keyway and key and subjecting the key to a severe crushing 
action rather than simple shear. If the conditions of service 
require a continual reversal of motion a state similar to tnat 
shown in Fig. 56 (c) is induced, where trie resisting forces F 3 
and F 4 have been moved inward and their moment arm made so 
short that their magnitude must be very great to hold the key in 
position. This may bring a severe bursting stress on the hub. 
It is evident, therefore, that keys which fit sidewise only, cannot 
be depended on to carry as great a load as those which fit well 
on the top and bottom. Where great accuracy is required, as 
13 



194 MACHINE DESIGN 

in machine tool construction, the hub is often made a force fit 
on the shaft and the key fitted only on the sides, so that it cannot 
throw the parts out of relative alignment by radial pressure. 
Referring to Fig. 56 (a) 
Let I = the length of the key or hub 
" /= " thickness of the key 
" b = " breadth of the key 
" T= " torsional moment applied to the shaft 
" P= " force acting at the radius of the shaft so that 
d 
2 
Then for shearing stress p s 

P = P.lb (1) 

and since the torsional moment applied to the shaft must equal 
the moment of the crushing load applied to the side of the key 

d 

2 

orP- =£/-(- ) .... (2) 

2 rc 2 V2 4/ w 

If F t be considered to act at the radius of the shaft (which can 
be done without serious error for keys as ordinarily proportioned) 
equation (2) reduces to 

p = pjt. ...... ( 3 ) 

Equations (1) and (3) may be used to compute the stresses in any 
sunk key. 

If the shearing resistance of the key is to equal the crushing 
resistance, then from (1) and (3) 

. * A . ; 2 p a 

. • 7 = — - or t = b— - (4) 

If p c = 2 p B , t = b, and the key is square for equal resistance to 
shearing ano^ crushing. For machinery steel, such as is generally 

A 
used in keys, — = .8, and hence from (4) for equal strength in 

re 



KEYS, COTTERS, AND FORCE FITS 195 

shearing and compression t = i.6b. If, in addition, the moment 

of the shearing resistance of the key is to be equal to the torsional 

resisting moment of the shaft; then 

d nd? 

T = pjb- = p>— . ... . ( S ) 

where p\ is the shearing stress in the outer fibre of the shaft. 
For steel shafts and keys, which are most common, p a = p\ whence 
from (5) 

»£-£ (6) 

2 10 

The minimum length of hub (/), as determined by practice, which 
is necessary to give a good grip on the shaft, should not be less 

7 d 
than — . Substituting this value of I in equation (6) 
2 

$bd 2 *d? 
4 16 

7T d 

. * . b = — d = - nearly (7) 

12 4 ' 

d 
The above would, therefore, give keys of breadth b = — , depth 

4 

or thickness t = 1.6 b = .4 d, and minimum length — d. Keys as 

used in practice conform closely to these rules as far as length 
and breadth are concerned; but, to avoid cutting away so much 
of the shaft, the thickness is usually much less than that given 

above. An average value of the thickness may be taken at — b. 

o 

This gives a key considerably thinner than it is wide and makes 
it weakest in crushing. The crushing resistance can, however, 
be increased by lengthening the key or by using a hard grade of 
steel. 

Keys designed as above usually have an excess of strength, 
since the friction between the shaft and the hub materially de- 
creases the load actually brought upon the key. In addition, as 
has been pointed out, shafts are most usually designed for stiff- 
ness or angular distortion, and therefore are greater in diameter 



196 



MACHINE DESIGN 



than would be required for strength alone. If the key is made 

proportional to the shaft diameter as above, it must, therefore, 

have excess strength against rupture; and such keys seldom fail 

unless subjected to severe shock or extraordinary loads. 

There are no fixed standards for the dimensions of keys, 

various machine builders having their own standards.* The 

following table may be taken as representing average practice 

•2 
when the length of the key is not less than - d. If the length 

must be less than this value, the crushing stress should be com- 
puted, as it may be necessary to use two keys. 



TABLE XI 

DIMENSIONS OF FLAT KEYS IN INCHES 



Diam. of Shaft d. . . . 


1 


i| 


il 


x 4 


2 


2h 


3 


31 


4 


5 


6 


7 


8 


9 


10 


Breadth of Key b. . . . 


1 
I 


5 

T6 


1 


7 
T6 


1 
1. 


5 
8 


3 

4 


8 


1 


ii 


t2 


1* 




2 


2i 


Thickness of Key t. . 


5 
32 


3 
T6 


1 

4 


9 

32 


5 


• 


7 

T6 


1 
% 


5 

8 


tt 


13 

T^ 


7 

8 


I 


ii 


II 



The taper of sunk keys is usually about yi" per foot of length. 

Another form of sunk key is shown in Fig. 58. This key 
drives by compression or as a strut. The keyways are more 
difficult to cut, the keys more difficult to fit, and the shaft is cut 
deeper than for the common form. It has been used with great 
success on very heavy work. 

70. Feathers or Splines. Sometimes it is desirable to have 
the hub free to slide axially along the shaft, but constrained to 
rotate with it. In such cases a feather or spline is used. The 
sides of the spline are parallel and it may be either fastened 
rigidly to the shaft or it may move with the hub. Small splines 
are frequently dovetailed into the shaft (or hub), as shown in 
Fig. 59 (a), while larger ones are often held in place by means 
of countersunk screws (Fig. 59 b), or rivets. 

A common way of securing the feather so that it will move 

* See Kent's "Mechanical Engineer's Handbook," page 977. 



KEYS, COTTERS, AND FORCE FITS 



x 97 



with the hub is shown in Fig. 60. Splines are subjected to a 
shearing stress across the mid-section at the radius of the shaft, 
and to a crushing stress on the sides in the same way as sunk keys. 
Being fitted loosely on the top and bottom, they do not produce 
any friction between the hub and the shaft and, therefore, offer 
much less resistance than sunk keys to the rolling action im- 
posed upon them (see Art. 69). This rolling action tends to 
bring a concentrated crushing force at a and b (Fig. 59 a), if the 




Fig. 58. 



Fig. 59. 



feather is not rigidly secured to either the hub or the shaft. For 
this reason, and in order also to provide ample wearing surfaces, 
feathers are usually given a greater radial depth than sunk keys, 
and from their general proportions are often distinguished as 
square keys. It is evident that the holding power of splines is 
not equal to that of sunk keys. 

The following table gives dimensions of feathers which agree 
with common practice : 

TABLE XII 

DIMENSIONS OF FEATHER KEYS IN INCHES 



Diam. of Shaft d. . . . 


1 


ii 


Ij 


T 3 - 


2 


2§ 


3 


3* 


4 


5 


6 


7 


8 9 


10 


Breadth of Feather b. 


1 
4 


5 

16 


3 

~5 


1 
16 


1 
2 


5 

8 


3 
4 


7 

8 


1 


i£ 


,3 
l 8 


li 


if J 2 


2| 


Thickness of Feather / 


I 


7 
1 


\ 


9 

1G 


5 
8 


3 
4 


8 


I 


i\ 


if 


If 


1* 


X\ 


2| 



The length of feather keys is, in general, greater than that of 
sunk keys, for the same size of shaft, in order to reduce the bear- 
ing pressure and increase the wearing surface on the sides. 



198 



MACHINE DESIGN 



71. Cotters. A cotter is a form of key used to prevent relative 
sliding between two members. Fig. 61 shows a method of 
securing a piston rod to a piston by means of a cotter. In this 
case the connection is permanent in character, the cotter being 
removed only when the piston or piston rod is repaired or re- 
newed. In other forms of cottered joints of this character the 
rod is not tapered, but is prevented from sliding into the boss by 
means of a shoulder or by the cotter alone. The cotter is usually 
rectangular in section, but sometimes the edges are rounded so 
as to avoid sharp corners in the opening cut through the rod or 
to facilitate machining. In light work a taper pin of circular 
section is often used as a cotter. Fig.* 62 shows an arrange- 




Fig. 60. 



Fig. 61. 



Fig. 62. 



ment of a gib and cotter (commonly known as a gib and key), 
such as is used on the ends of the connecting-rod of steam engines. 
The function of the gib is to prevent spreading of the strap. 
This arrangement permits a small amount of adjustment between 
the strap and the connecting-rod for taking up wear on the pin 
and brasses. 

72. Stresses in Cotters. A cotter of the form shown in Fig. 
61 is a beam supported at the ends. The exact distribution 
of the loading is indeterminate, as the bending of the cotter tends 
to concentrate the load near the points of support. It is suffi- 
ciently accurate, however, to consider the load as uniformly dis- 
tributed. The area of the surface of the cotter where it bears 
on the rod, and also on the hub, should be sufficiently great to 
prevent crushing of the material. This indicates that the diam 



KEYS, COTTERS, AND FORCE FITS 1 99 

eter of the hub should, for similar materials, be twice that of 
the rod, which is the usual proportion. The section of the cotter 
at the point of support should be great enough to prevent shearing, 
and in many cases it is sufficient to compute the section for shear 
alone, neglecting the bending action. 

When a cottered joint of this character is made, the cotter 
must be driven in tight enough to prevent its backing out. This 
is especially true when the load is a reversed one as in the case 
of the steam-engine piston. This induces an initial stress in the 
cotter and rod, over and above that due to the load P. The 
conditions, in fact, are somewhat similar to those which exist in 
screwed fastenings (see Art. 60). The initial stress due to the 
driving of the cotter cannot be accurately computed, though it 
may be very great. For this reason all calculations of dimensions 
based on the maximum applied load should be modified to suit 
the conditions of service and the materials of which the joint is 
made. Thus if the rod be of brass and the hub or boss of steel, 
as is common in pump work, the proportions would be different 
from those employed if all the materials were of steel or of steel 
and cast iron. 

Let d = the diameter of the rod where the cotter passes through. 

" / = thickness of cotter. 

" b = breadth of cotter. 
Then, in order that the net cross-section of the rod may be as 
strong in tension as the cotter and rod, where they bear upon 
each other, are in crushing, 



A^-td) = td P. 



••"■^1) W 

For a steel rod and steel cotter where p c = .8 p t , t = .44^ . (2) 

Good practice gives 6 = 4/ = 1.76 d (3) 

The taper of cotters, as shown in Fig. 61, should be so small 
that there is no danger of backing out and should not exceed ]/2 
inch per foot of length. An auxiliary locking device is often 



200 MACHINE DESIGN 

used in arrangements such as shown in Fig. 62, in which case 
the taper may be as great as 1 in 8. 

In the form of cotter shown in Fig. 62, the stress due to 
driving the key may be disregarded, and the design based on the 
maximum applied load. The student is referred to treatises on 
steam-engine design for relative proportions of this form. 

It is often necessary to allow a rather high bearing pressure 
on the cotter to avoid large and clumsy proportions. An ex- 
amination of successful practice shows an allowable pressure of 
15,000 pounds per square inch as computed from the applied 
load. 

FORCE AND SHRINKAGE FITS 

73. General Considerations. Crank discs, the hubs of heavy 
fly-wheels, impulse water wheels, and work in general which is 
to be subjected to shock or vibration, must be fastened to the 
shaft more securely than can be accomplished with a key, when 
the hub is a sliding fit on the shaft. In such cases the bore of 
the hub is made slightly smaller than the diameter of the shaft, 
and the shaft is forced cold into the hub ; or the hub is expanded 
by heating till the bore is slightly larger than the shaft, then 
slipped over the shaft and allowed to cool in place. The first 
method is known as a force or pressure fit, and the second as a 
shrinkage fit. The degree of tightness or "grip" required be- 
tween shaft and hub depends largely on the service. Thus, with 
shafts up to three or four inches in diameter, a difference between 
the diameter of the shaft and the bore such that the parts may 
be driven together with a hand sledge is often satisfactory. Such 
a fit is called a driving fit, and the difference between the shaft 
diameter and the bore is very small. With such work as arma- 
ture spiders and fly-wheel hubs, the allowance for the press fit 
depends largely on the facilities for erection. If the parts can be 
forced together in the shop, where adequate means, in the form 
of a powerful hydraulic press is to be had, an allowance requiring 
a pressure of one hundred tons or more may be made. But if 
the parts must be erected in the field, this allowance may have 
to be reduced on account of the difficulties of erection. It is 



KEYS, COTTERS, AND FORCE FITS 201 

usually possible in the case of armature spiders, fly-wheel hubs, 
etc., to obtain a sufficiently tight grip on the shaft by means of 
a press fit without inducing undue stress in the parts. Depend- 
ence for preventing relative rotation may be, in a large measure, 
placed upon the key in all such cases. 

In such work as crank shafts when built up from separate 
parts, it is often necessary to insure as strong a grip upon the 
shaft as is possible without inducing undue stress. A greater 
difference between the shaft diameter and the bore of the hub is 
then allowed than in forced fits and the parts are usually put 
together by shrinking. In the latter cases the stresses induced 
are of importance and should be carefully considered. 

74. Stresses Due to Force Fits. If x be the elongation or con- 
traction of any radius r, then 2 x % is the corresponding elongation 
or contraction of the circumference 2 w r. The elongation or 

2 n x 
contraction of the circumference per unit of length is . If 

2 7r r 

p be the stress which would induce this change of length of 
circumference, and E be the coefficient of elasticity of the material, 
then 

E = -1— or x = ^ (1) 

2 *x E v J 

2 ~ r 

In Fig. 63, let A represent a hollow shaft on which has been 
forced or shrunk a hub or boss B, the radius of the contact surface 
being r 2 . Before the operation of pressing, the outer radius of 
the shaft was r 2 + e 2 , and the inner radius of the hub was r 2 — e' 2 . 
The hub B is, therefore, in the condition of a thick cylinder sub- 
jected to an internal pressure, and the shaft A is in the condition 
of a thick cylinder subjected to an external pressure. The great- 
est tensile stress will be found at the inside surface of the hub, 
and the greatest compressive stress at the inside surface of the 
shaft. If, therefore, e be the difference between the outer radius 
of the shaft and the inner radius of the hub, before pressing, 
then e = e 2 + e' 2 . 



202 MACHINE DESIGN 

Let p t be the unit tensile stress in the hub at a radius r 2 . 
Let p c be the unit compressive stress in the shaft at a radius r 2 . 
Let w 2 be the unit radial pressure between A and B. 
Let r t be the internal radius of the shaft. 
Let r 3 be the external radius of the hub. 
Then from (i) 

A p c Ee 

6 = E r * + E r * ° r Pt + Pc = 7" ' * * ^ 

The general equation for the stress in thick cylinders of this kind 
is,— 

2 r{ w i — 2 r 2 z w 2 + — -^ — (w l - w 2 ) 
* P= 3 W ~ r>) • • • (3) 

Where r t is the inner radius of the cylinder, r 2 the outer radius, 
w t the internal unit pressure, w 2 the external unit pressure and 
p the tensile or compressive stress at any radius r. Applying this 
equation to the shaft, w 1 = o, r = r 2} whence the compressive stress 
at the surface of the shaft is 

w 2 (2 r 2 2 + 4 r 2 ) 

*•" jW-O =aW >- ■ ■ ■ (4) 

In a similar way substituting r 2 for r h r s for r 2 , w 2 for wi, 
and Wz for w 2 , the unit tensile stress on the inner surface of 
the hub is 

w 2 (2 r 2 2 + 3 r, 2 ) 

A " 3W-0 -' w " • • • (s) 

Dividing (4) by (5) 

Pc a 



From (2) and (6) 



(6) 

Pt 

Eep 
Pi r 2 ( « + p) W 

Ee a 

* The following treatment is from Professor Merriman's " Mechanics of 
Materials," 1906 edition, page 396. The notation has been changed to agree with 
that adopted in this work. 



KEYS, COTTERS, AND FORCE FITS 203 

When the shaft is solid, r x in the above equation becomes zero 
and the equations are much simplified. 

Example. A hollow steel shaft 10 inches outside diameter 
and 2 inches inside diameter is to have a steel crank shrunk upon 
its end. The hub of the crank is 18 inches in diameter. What 
must be the difference between the diameter of the shaft and the 
bore of the crank so that the tensile stress at the inner surface 
of the hub shall not exceed 20,000 lbs. per square inch? What 
will be the corresponding compressive stresses at the outer and 
inner surfaces of the shaft ? Take E = 30,000,000. 

Heref^ = 1, r 2 = 5, r 3 = 9 and p t = 20,000 

2^ + 4^ (2 X 5 2 ) + (4 X i 2 ) 3 



Whence a = 



3(^ 2 -0 3(5 2 -i 2 ) 



2r 2 2 + 4 r 3 2 (2 X 5 2 ) + (4 X 9 2 ) 

and/9 = —7-= ^ = 71 =: =2.23 

3W-'2 2 ) 3(9— 5 2 ) 



Then from (7) 



Af, (« + fl 2 °>°°°x 5 (J+2. 23 ) 

e = = ; = .0044 

h, /? 30,000,000 X 2.23 



From (8) 














P." 


Ee 


a 


3° 


,000,000 X 


.0044 


X . 


r 2 ( a 


+ P) 




5 X: 


2.98 




From (4) 


















W 2 


a 


6,700 
1 


8,900 


lbs. 



75 



= 6,700 



and substituting this value in (3), making r = r l and w 1 =o, it is 
found that the compressive stress at the inner surface of the shaft 
is 18,500 lbs. per square inch. 

It is evident that if e be assumed, which is usually the case, 
the resulting pressure and stresses can be computed. It should 
be noted that p t must be well within the elastic limit to prevent 
the hub yielding and relieving the pressure. It appears, as 
pointed out by Professor Merriman, that the allowances made in 
practice for force fits, induce stresses which should be considered 



204 



MACHINE DESIGN 



if other stresses are to act on the members. Thus, in the example 

given, the total allowance or difference between the diameter of 

the shaft and the bore of the hub would be 2 X .0044 = .0088 ; 

.0088 

and the allowance per inch of diameter would be = .00088" 

10 

which is close to average practice for force fits, where .001 " per 
inch of diameter is often allowed. A somewhat greater allowance 




Fig. 64. 



Fig. 65. 



is generally made for shrinkage fits, as here the difficulty of forcing 
on the hub does not occur. 

75. Practical Considerations in Force and Shrink Fits. The 
foregoing equations, while giving the probable stresses and radial 
pressure resulting from a force or shrink fit made with an allow- 
ance e, are limited in their application to the practical making of 
jorce fits. There is, generally speaking, no difficulty in making 
shrink fits, with any practical allowance, as far as getting the 
parts together is concerned; although greater skill is required in 
handling shrink fits than force fits. In making force fits, how- 
ever, the amount of pressure that can be applied to the parts 
is often a controlling factor. The probable radial pressure be- 
tween the shaft and hub (w 2 ) may be found as above, but little is 
known of the coefficient of friction in such work, and it is evident 
that this quantity will vary greatly with the character of the 
material, the finish of the surface, and the lubricant applied. 
Experimental data are lacking on this point, hence it is almost im- 



KEYS, COTTERS, AND FORCE FITS 205 

possible to estimate the resistance to slipping offered by force or 
shrink fits. In general, shrink fits are superior to force fits 
since their surfaces are very dry and unlubricated, while those 
of a force fit are lubricated. Total dependence is, therefore, 
seldom placed on the fit itself, but a key is also used for safety. 
Experience shows that the pressure required to make a force 
fit will vary for any given diameter. 

(a) Directly as the length of the hub 

(b) Directly as the allowance e 

(c) As some function of the radial thickness of hub 

(d) With the character of the materials and the finish of 

the surfaces. 
It is evident that a mathematical expression accurately expressing 
these relations would not be practicable, and recourse must be 
had to successful practice. 

An allowance of .001 " per inch of diameter will represent 
average practice in this country for such work as crank shafts, 
crank pins, and in general where a tight fit is required. For 
armature spiders, or fly-wheels, one-half this allowance is often 
sufficient. For shrink fits a greater allowance is often made, 
although the foregoing discussion indicates that this should not 
be much exceeded considering the stresses induced. 

For further information and practical data the student is 
referred to the following : 

Transactions of the American Society of Mechanical En- 
gineers, Vol. XXIV. 

"Machine Design" by Forrest R. Jones. 

"Machine Design" by W. L. Cathcart. 

Machinery, Vol. Ill, No. 9, May, 1897. 

76. Thin Bands or Hoops. If the ring or band which is forced 
or shrunk on to a member be thin, radially, compared to its 
diameter, the assumption can be made, without appreciable error, 
that the stress is uniform throughout the cross-section of the ring. 
The change of form in the member on which the band is placed 
due to compression is so small in such cases that it may be 
neglected, and the stress in the band may be taken as that due to 
stretching it over an incompressible body. This is practically 



206 MACHINE DESIGN 

applicable to any ordinary shape of band, but rigidly true for 
circular shapes only. Thin bands of this character are usually 
shrunk into position. 

Example. A thin steel band is to be shrunk on to a casting 
whose external linear dimension where the band is to be placed is 
48 inches. What must be the length of the inside face of the band 
so that the stress per unit area due to shrinking will be 30,000 lbs. ? 
What will be the area of the cross-section of the band in order 
that the total stress in the band may be 60,000 lbs. ? 

Let / = the length of band before shrinking. 

Then 48 — / = total amount of elongation of band. 

48 — / 
and — - — = unit elongation of band. 

Whence, if E, the coefficient of elasticity, be taken as 30,000,000, 

. ^ unit stress 30,000 

then, E = 30,000,000 = unitstrain = ^7, or / = 47-95 in* 

/ 

60 OOO 

The total area of the cross-section of the band will be A = — - 

30,000 

= 2 square inches which may be distributed in any convenient 

proportions. 

If the part on which the band is to be shrunk is circular in 
form, the band is in the condition of a thin cylinder subjected to 
an internal pressure w per unit area, where w is the radial pressure 
between the band and the part on which it is shrunk. 

Therefore by Art. 78, wd = 2 P, where P is the total stress 

2? 

per unit width of the band or w = —7-. Thus in the above prob- 

j.8 
lem let the band be shrunk upon a circular hub of diameter — , 

and let the cross-section of the band be X" x 4". Then P = 

60,000 2 P 2 X 15,000 , 

= 15,000, and w = —j- = q = 1,962 lbs. per 

4 a 48 

square inch. ~ 

The steel tires of locomotive driving wheels are usually shrunk 
on with an allowance for shrinkage of .ooi // per inch of diameter 
which gives .001 inches elongation per inch of circumference. 



KEYS, COTTERS, AND FORCE FITS 207 

Taking £ = 30,000,000, and considering the tire a thin band, the 
unit stress in the tire is 

p — E A = 30,000,000 X .001 = 30,000 lbs. 

77. Other Forms of Shrink Fits. Many machine parts such 
as fly-wheel rims are held together by steel links or bands shrunk 
into place. The theory outlined in the preceding article is clearly 
applicable to these members, and their dimensions should be 
carefully calculated so that they will not be overstrained by the 
shrinking alone. If such members are so designed that they will 
be stressed up to the elastic limit from shrinkage alone, they are 
liable to be strained beyond the elastic limit, when an external 
load greater than the total shrinkage stress is applied to the parts 
which they hold together, and the link, taking a permanent set, 
becomes ineffective. In computing the dimensions of such links 
allowance must sometimes be made for the compression of the 
parts held together, but ordinarily this is small and may be 
neglected. 

Occasionally a bolt or link is used to reinforce a cast-iron 
member against tensile stress. Thus in open frames, Fig. 65, a 
large bolt is sometimes placed on each side of the throat as 
shown. These bolts are usually put in hot and allowed to cool in 
place. As ordinarily applied, the benefit derived from them is 
questionable. If they are designed and fitted so as to put the 
frame in compression at A , an amount equal to the tension induced 
by the working load P at this same point, without being themselves 
strained beyond the elastic limit when the load is applied, then 
no stress can come upon the frame itself from the force P. If, 
however, the bolts and frame are each to carry part of the load, 
care should be exercised that the stress induced in the bolts by 
the initial load due to shrinking is so low that the additional 
stress due to the external load does not raise this initial stress 
beyond the elastic limit, thus giving the bolts a permanent set 
and destroying their usefulness. 

Let A, Fig. 64, represent a cast-iron member of uniform 
cross-section which is to be reinforced against tensile stress by the 
bolt B. Suppose, first, that the nut is screwed up till it just 



208 MACHINE DESIGN 

bears firmly on the casting. If now an external tensile load is 
applied to the casting, the bolt and casting will be elongated 
the same amount A. But the coefficient of elasticity of cast 
iron is only about one-half that of steel. Hence, since p = EA, 
the stress per unit area in the casting will only be one-half that 
in the steel. If 2,000 lbs. is the allowable unit stress in the 
casting, 4,000 lbs. per unit area is all that can be thus obtained 
in the bolt. This would lead to unnecessarily large bolts. 

Suppose, however, that the nut is set up till a total compres- 
sive load W is applied to the cast iron. The bolt will be elongat- 
ed* and the casting compressed, the amount of elongation or 
compression depending on the cross-section of the respective 
members. The unit stress induced in the bolt and casting will 
also be proportional to the area of their respective cross-sections. 
If now an external tensile load W is applied to the bolt, the 
tendency is to relieve the compressive stress in the casting and to 
increase the tensile stress in the bolt. When the load applied is 
sufficient to elongate the bolt as much as the casting was origin- 
ally compressed, the casting will be relieved of all stress. If the 
external load W is applied to the bolt through the casting itself, 
it is evident that practically the same result is obtained; and after 
the compressive stress in the casting is fully relieved any further 
addition to W induces a tensile stress in the casting and still fur- 
ther increases the tension in the bolt. Usually the cross-sectional 
area of the casting is very much greater than that of the bolt. 
Furthermore the compressive stress induced in the casting by the 
initial load on the bolt is usually very small compared to the 
tensile stress induced by the working load. For these reasons 
the compressive deformation in the casting can usually be 
neglected without appreciable error; and the bolt may be 
designed on the basis of the external load alone. (See Art. 
60, Case a.) 

Example. In Fig. 65 let the section AB be stressed by the 
load P whose arm is /. Let O be the location of the gravity 
axis of the section AB. It is desired to keep the stress at A not 

* See Art. 59. 



KEYS, COTTERS, AND FORCE FITS 209 

greater than 3,000 lbs. per square inch. The material is to be 
cast iron. Let P = 60,000. 

" ' / = moment of inertia of section = 4,500. 

" e = 10 inches. 
Also let the area of the section be 200 square inches. Then from 

(P Ple\ 

(M), page 94, the tensile stress at A due to P is p = ( — + ~j~) 

( 60,000 60,000 X 30 X io \ . , . , 

= I + J = 4,300 lbs., and it is desired 

V 200 4,500 / ° 

to reduce this to 3,000 lbs. by reinforcing bolts. These rein- 
forcing bolts serve the double purpose of increasing the factor 
of safety by reducing the fibre stress, and also of decreasing the 
deflection of the frame at the point where the work is done. Let 
these bolts be located 8" from O. Then the compressive stress 
induced at A by P' is 

(F P' V e \ (P^_ P'X8 X io \ 
\4 I ' ^200 4,500 / 

But p — p' must equal 3,000; therefore 

/ P' P' X 8\ 

4,300 — I + ) = 3,000. 

^ ,0 V200 450 / °' 

Whence P' = 57,000. This is the total tensile load on both bolts, 
when the full working load P is applied. If the maximum stress 
at the root of the thread be taken at 15,000 lbs., then the area of 

each bolt at the root of the thread is = 1.0 so. in., 

2 x 15,000 ^ 

which corresponds closely to a i%" bolt. The area of the 

body of a i%" bolt, where most of the stretching takes place, 

is 2.4 square inches. Hence the working stress in the body of the 

57,000 

bolt is = 11,880 lbs. per sq. in. That portion of the 

2x2.4 

boss which immediately adjoins the throat is subjected to an aver- 
age tensile stress nearly equal to the fibre stress at the surface 
of the throat or 3,000 lbs. per square inch. The upper and lower 
portions of the boss have little or no tensile stress induced in 
them, as a consideration of a section such as XX whose gravity 
axis is at O', will show. It will be reasonable to estimate that 
14 



2IO MACHINE DESIGN 

the stress in the boss is equivalent to the full stress of 3,000 lbs. 
per square inch through 14 inches of its length, the total length 
being 19". Neglecting the compressive deformation of the boss 
due to the initial load from the bolts, the stress induced in the 
bolt when the stress in the boss is 3,000 lbs., will be 3,000 
X2X 14/19 = 4,500 lbs. (remembering that the coefficient of 
elasticity of steel is twice that of cast iron). Whence the 
initial stress in the bolt will be 11,880 — 4,500 = 7,380. The 
allowance for shrinkage necessary to give this initial stress will be 

A - l 4 - ±9X^8? = 6 „ 

h, 30,000,000 

The number of threads perinchon ai^" bolt is 5. Hence after 

.0046 
the nut has been set up snugly it should be given = .023 

5 
of a turn, or should be turned through 360 X .023=8.2 degrees. 
This is most easily, done in the case of large bolts by first marking 
the nut with reference to the bolt when set up snug in a cold state, 
and then heating the body of the bolt, if necessary, and rotating 
the nut the desired amount, allowing it to cool in position. 

It is to be especially noted that a very small shrinkage allow- 
ance is needed to induce a great stress in the bolt. If too great 
an allowance is made, the bolts may be stressed beyond the elastic 
limit, and take permanent set the first time the external load 
is applied. When the external load is again applied, a force much 
smaller than the total load P will strain the casting to the 
point where the bolt becomes effective. The total load P will 
strain the casting further than it did originally, and even if 
the stresses induced are not sufficient to rupture the casting, the 
stiffness of the frame is materially decreased. 



CHAPTER IX 

TUBES, PIPES, CYLINDERS, FLUES, AND THIN PLATES 

78. Resistance of Thin Cylinders to Internal Pressure. If a 
hollow circular cylinder, whose walls are very thin compared to 
its diameter, is subjected to an internal bursting pressure, a 
tensile stress is induced in the walls. This tensile stress is reduced 
near the ends by the action of the ends themselves which tend to 
hold the walls together. Let Fig. 66 represent one-half of a por- 
tion of a thin cylinder so far removed from the ends that their 
effect may be neglected. 

Let w = the unit internal pressure 
d = the diameter of the cylinder 
r = the radius of the cylinder 
/ = the thickness of the cylinder walls 
p = the unit tensile stress in the longitudinal section 
p t = the unit tensile stress in the transverse section 
I =the length of the part considered 
Consider the half of the cylinder as a free body, and resolve all 
forces perpendicular to the cutting plane. The normal pressure 
on a longitudinal strip of length / and width rdo is w I r do. The 
component of this force perpendicular to the cutting plane is 
wlr dd sin 0. The total pressure normal to this plane is 

J wlr do sin = w I r J sin odo = 2wlr = wld. 

For equilibrium this normal force must equal the resisting stress 
in the two sides of the cylinder. Hence 
2 pit = wl d 

w d , x 2 pi , x w d , s 
or p = (1) ; w = — r - (2) ; or t = (2) 

In other words, the unit longitudinal stress in the walls of a thin 

211 



212 



MACHINE DESIGN 



cylinder is equal to the product of the diameter into the unit inter- 
nal pressure, divided by twice the thickness of the cylinder walls, 
and is independent of the length of the cylinder. 






Fig. 66. 



Fig. 67. 



If a transverse section of the cylinder (Fig. 67) be considered, 
it will be seen that the total pressure on the head, which tends 



to cause rupture along a transverse section, is 



7T d 2 w 



and 



this must be equal to the intensity of the transverse stress pro- 
duced multiplied by the area of the metal in such a section, or, 

— — = r:dtp t 



* 



A = 



wd 



(4) 



w = 



4PJ 



(5) 



or / 



wd 
4 A 



(6) 



\t d 

A comparison of (1) and (4) shows the stress in transverse sections 
to be only one-half of that in longitudinal sections. For this 
reason it is very common practice to make the circumferential 
seams of a boiler shell with a single riveted joint, when the 
longitudinal seams are double or triple riveted. 



* In this discussion the mutual interaction of the longitudinal and transverse 
stresses is neglected. If a tensile stress p% is induced in a body, the body contracts 
laterally as if acted upon by a stress "kp t acting at right angles to the line of action 
of pt where a is Poisson's ratio. (See Merriman's " Mechanics of Materials," 1906 
edition, page 359.) Therefore the true longitudinal stress p\ in the above case 
(since a equals J for steel) is 

1 w d 1 w d w d 

*_,__*_____ .8 S _. 

This gives a lower value than equation (1) and hence the latter is on the side of 
safety. 



TUBES, PIPES, CYLINDERS, FLUES, AND THIN PLATES 213 

79. Thin Spheres. Since all the meridian sections of a sphere 
are the same as the transverse section of a cylinder of equal 
diameter, it is evident that the stress in the walls of a sphere is 
given by (4). If spherical heads, of the same thickness as the 
shell, are placed on a cylinder which is to withstand internal 
pressure, they will be subjected to a maximum stress equal to the 
transverse stress in the shell. 

80. Resistance of Non-Circular Thin Cylinders to Internal 
Pressure. Suppose a cylinder to have a cross-section made up of 
circular arcs as in Fig. 68. Take the upper half as a free body 
(section along the major axis). Let the resultants of the com- 
ponents 'of pressure which are normal to the plane of the section 
be W 1} W 2 , W 3 , for the portion marked I, II, III, respectively. 
Then these resultant forces per unit of length of the cylinder are 
as follows: — 



W \ = w r I sin <p d <p = w r ( — cos<p + cos 0) 



w m< 



sin d = w R ( — cos B f + cos 0") = w m 2 

W 3 = w r J sin <p d <p — wr ( — cos n + cos <p") = w m z 

Therefore W ± + W 2 + W 3 = w {m x + m 2 + ni 3 ) = w A 

In a similar way, if the section is taken along the minor axis, 
the resultant force normal to this axis is found to be wB. In like 
manner the resultant force normal to any section is (per unit of 
length of cylinder) equal to the intensity of pressure multiplied 
by the axis of that section. As B is less than A, the resultant 
force wB is less than wA; or the force tending to elongate the 
minor axis is greater than the force tending to elongate the major 
axis. If the tube were perfectly flexible, its form of cross-section 
would become, under pressure, one in which all axes are equal, 
or circular. A rigid material offers resistance to such change of 
form, and a flexural stress is produced in addition to the direct 
tension, but it approaches nearer to the circular form as the pres- 
sure increases. The existence of this flexural stress in a non- 
circular cylinder becomes apparent from a comparison of Figs. 69 



214 



MACHINE DESIGN 



and 70. In Fig. 69 (circular section) the lines of normal pressure 
all pass through a single point (the centre of the circle) ; the re- 
sultant (P r ) of the tensions (P t and P 2 ) also passes through this 
same point, hence these forces form a concurrent system, and 
they are in equilibrium. In Fig. 70, however, the pressures do 
not in themselves form a concurrent, nor parallel, system of forces, 
hence they cannot be balanced by a single force (as the resultant 
P r ), but there must be a moment, or moments, of stress for equi- 
librium. A similar course of reasoning could be applied to a 
cylinder of any non-circular cross-section; for such a section 
(Fig. 71) could be considered as made up of circular arcs, each of 
which could be treated (like the special case of Fig. 68) by inte- 





Fig. 68. 



Fig. 71. 



grating between proper limits. A direct inspection will also show 
that in any non-circular section cylinder, subjected to internal 
pressure, the pressure tends to reduce the cylinder to a circular 
cross-section. Suppose the original cylinder (Fig. 71) to be cut 
along the greatest axis of its cross-section, and that a flat bottom 
coinciding with this section-plane be secured to it, the lower 
portion of the cylinder being entirely removed. The total 
pressure on this bottom evidently balances the components of 
the pressure on the curved surface which lie normally to this flat 
bottom ; hence, the resultant of these normal components of pres- 
sure equals w (a . . a) =w A, per unit of length of cylinder. 
In a similar way, the resultant of components of pressure acting 
normally to any other section (as b . . b, Fig. 71) equals 



TUBES, PIPES, CYLINDERS, FLUES, AND THIN PLATES 215 

w (b . . b) —w B < w A. This direct method might have 
been used in the preceding cases (Figs. 66 and 68) without re- 
course to the calculus. 

It is apparent, then, that any cylinder under internal pressure 
tends to assume a circular cross-section. A cylinder of nominal 
circular section, but departing from the true form to some extent, 
tends to correct this departure under internal pressure; or if a 
circular cylinder under internal pressure is deformed by any ex- 
ternal force, it tends to resume its circular shape. Thus a circular 
cylinder under internal pressure is in "stable equilibrium." If 
the section is other than a true circle there is a flexural stress, as 
well as tension, when under pressure. 

RESISTANCE OF THIN CYLINDERS TO EXTERNAL 

PRESSURE 

81. Theoretical Considerations. If a thin hollow cylinder of 

circular section is subjected to an external pressure, it is obvious 

that a course of reasoning similar to that in Art. 78 will show 

that a compressive stress is induced in the walls of the cylinder, 

the value of which will be given by formula (1) Art. 78 or 

wd 
p — where p is a compressive stress. 

If the cylinder were perfectly cylindrical, of uniform thickness, 

and of homogeneous material, there seems to be no reason why 

failure should occur until the compressive stress reaches the yield 

point of the material. But tubes are never absolutely circular in 

form, uniform in thickness, or homogeneous in character; and 

hence failure occurs long before the compressive yield point is 

reached. A tube which fails under external pressure is said to 

collapse, and the forms of collapsed tubes are very characteristic. 

Fig. 72 shows the form of cross-section of collapsed tubes, and 

Unwin* has shown that the number of lobes depends on the 

ratio of length to diameter, the smaller this ratio the greater being 

the number of lobes. This peculiarity is undoubtedly due to the 

/ 
influence of the heads placed in the ends. For values of -r 

* See " Elements of Machine Design," page 101, 1901 edition. 



2l6 



MACHINE DESIGN 



greater than about 4 to 6, only the forms of collapse shown at 
c and d, Fig. 72, appear. 

If the non-circular cylinders of either Fig. 68 or 71 be con- 
sidered as subjected to external pressure, the force tending to 
increase the major axis will be seen to be greater than that tend- 
ing to increase the minor axis; hence the external pressure will 
cause collapse, unless the flexural rigidity of the material is 
sufficient to prevent this action. In a cylinder of nominal cir- 
cular section any departure from the ideal section will be increased 
by the external pressure. Or, if a cylinder of true circular 
section is deformed in any way while under external pressure, 
this pressure will tend still further to increase the deformation. 




Fig. 72. 



Fig. 73. 



In other words, a cylinder under external pressure is in " unstable 
equilibrium." As perfectly true circular sections and homogene- 
ous materials are not attainable in practice the danger of collapse 
must be taken into consideration in designing pipes, tubes, or flues 
to withstand external fluid pressure. 

Since the wall of an ideal thin tube is subjected to a uniform 
compressive stress, it may be considered as being in the same 
condition as a long column; and theoretical equations expressing 
the relation between the external pressure, the stress, and the 
dimensions of the tube have been developed on this basis. In 
view of the fact that the theory of long columns is itself most un- 
satisfactory, it is not surprising that such equations do not accord 
with actual results, and they may be safely disregarded, but the 
analogy between long compression members and tubes sub- 
jected to external pressure is instructive. Other deductions 
based upon the theory of elasticity, while throwing some light 



TUBES, PIPES, CYLINDERS, FLUES, AND THIN PLATES 21 7 

on the form of rational equations expressing these relations, are 
not as yet applicable to practical problems. 

82. Long Tubes, Pipes, etc. Until very recently the only 
experimental results on the collapse of thin tubes were those due 
to Sir William Fairbairn, who, in 1858, made a series of careful 
experiments on short tubes and deduced therefrom the following 
formula: 

/ 2.19 
w = 9,675,600 — (1) 

where w is the unit external collapsing pressure in pounds per 

square inch and t, I, and d are the thickness, length, and outside 

diameter respectively in inches. Fairbairn himself modified this 

equation, for simplicity, to the form 

t 2 
w = 9,675,600— (2) 

Many other equations have been deduced from the experiments of 
Fairbairn, usually of the same form but with different exponents. 
Thus Professor Unwin gives the following as the result of a careful 
resume of Fairbairn's work: 

For tubes with a longitudinal lap-joint 



2.21 



w = 7,363,000^^ (3) 



For tubes with a longitudinal butt-joint 

£2.21 

fid 



w = 9,614,000 Jn jii6 (4) 



For tubes with longitudinal and cross joints like an ordinary 
boiler flue 



2.3s 



w = I5j547t ooo—— 6 (5) 

Other writers have deduced similar equations from the same data. 
Fairbairn' s experiments were conducted with tubes whose 
lengths were small compared to their diameters. In such tubes 
the effect of the supporting action of the head is noticeable; hence 
his equations make the allowable pressure vary inversely as some 
function of the length. Now it is reasonable to suppose that if 



2l8 MACHINE DESIGN 

the tube were long enough the head would have no effect, except 

near the ends, and the collapsing pressure would be independent 

of the length. In a similar way if the tube were very short, the 

walls should theoretically yield by crushing, and the intensity of 

w d 
the compressive stress would be given by formula (i) or, p = • 

2 Tr 

In 1906 Professor A. P. Carman published* the results of a 
set of experiments made at the Engineering Experiment Sta- 
tion of the University of Illinois, which prove conclusively 
that Fairbairn's equations hold only for tubes whose lengths 
are from four to six times their diameters; and that be- 
yond that ratio the collapsing pressure is independent of 
the length. He found that the results of his experi- 
ments could not be well expressed by a single equation, but 
devised two equations to cover the range; these equations ex- 

t 
pressing the relation which exists between w and — . Thus for 

d 

values of — V025, and length greater than 4 to 6 times the 

t 
diameter, he gives w = k — — c where k and c are constants to 

a 

be determined experimentally and depending upon the material. 
For brass tubes 

i 
w = 93>3 6 5 ~ d ~ 2 A74 (6) 

For seamless drawn cold steel 

w = 95>5 2 ° ~ d ~ 2,090 (7) 

For lap- welded steel 

/ 
w = 83,270 - - 1,025 (8) 

Professor R. T. Stewart, f in an elaborate set of experiments 

* See Bulletin of the University of Illinois Engineering Experiment Station 
Vol. Ill, No. 17, June, 1906. 

f See Transactions of American Society of Mechanical Engineers, Vol. XXVII, 
1906. 



TUBES, PIPES, CYLINDERS, FLUES, AND THIN PLATES 2ig 
on lap-welded steel boiler tubes made for the National Tube 

t V 

Company, found that for values of — \ .023 the results of his 

work could be expressed by the following: 

t 
w = 86,670* — 1,386 (9) 

which corresponds closely with (8) of Professor Carman's work, 
showing the accuracy of the experimental work. 

For values of — <.o25 Professor Carman found that the 
a 

results of his work could be expressed by an equation of the form 

™ = k 'ii) 3 •■•••• do) 

where k r as before is an experimentally determined constant, 
whose value for thin brass tubes is 25,150,000, and for thin cold- 
drawn seamless steel tubes 50,200,000. 

/ 
Professor Stewart found that for values of — below .023, or 

practically the same limit as above, his results were expressed by 

w = 1,000 [1 - y 1 — l5 6oo-^-) . . . (11) 

The value of w for — = .023 is about 600 lbs., which corresponds 

closely with the upper limiting value of w obtained from (10). 

For values of - less than .023, the corresponding values of 

w, as found by either (10) or (11), do not differ materially. 

t y 
Furthermore, tubes in which — ^ .02 are not much used in en- 

d \ 

gineering work under external pressure, and for convenience 
therefore equation (10) will be adopted. 

83. Summary of Equations for Long Tubes. The works of 
Stewart and Carman deal entirely with tubes which are so long 
that the supporting effect of the heads is negligible, or in which the 
length is at least four times the diameter. Their experiments, while 



220 MACHINE DESIGN 

conducted separately, supplement and corroborate each other. 
As given above, the equations are not in the most convenient 
form for use by the designer, since usually /, d and w are known 
and / is required. Transposing these equations, therefore, they 
may be written as follows: 

For values of — <^ .023 and pressures less than 600 equation 



d 
10 becomes 

t = d 



Vf ™ 



where £ = 25,150,000 for thin brass tubes, and 50,200,000 for thin 
cold-drawn seamless tubes or lap-welded steel tubes. 

K 
For values of —\ .023 and pressures greater than 600 lbs., 

equation 6 becomes 

d(w + c) 
1 = k— (I3) 

where for brass tubes .... £ = 93,365 and c = 2,474 
" " seamless cold- 
drawn steel . . . £ = 95,520 and c = 2,090 
" lap- welded steel . . £ = 83,270 and c = 1,025 
The following approximate formula, which covers practically 

/ 
the whole range of values of — , is suggested by Professor Car- 
man as useful in making rough calculations. 

w = k" (|) 2 (13a) 

where k" = 1,000,000 for cold-drawn seamless tubes and 1,250,000 
for lap- welded steel tubes. From this, the following usually 
more convenient formula can be derived: 

< = <^f, ('3b) 

Example. A lap-welded steel boiler tube 4 inches outside 
diameter and 10 feet long, is subjected to an external pressure of 
300 pounds per square inch. What must the thickness be in 
order to have a factor of safety of at least 6 ? 



TUBES, PIPES, CYLINDERS, FLUES, AND THIN PLATES 221 

Here the assumed collapsing pressure is 

300 X 6 = 1,800 lbs. per square inch. 

Applying equation (13) 

d (w + c) 4(1,800+1,025) . 

t = = = .14 inch. 

k 83,270 

Here the ratio — = = .035, and hence equation (13) applies. 

d 4 

In case this ratio should be less than .023, which will seldom 
occur, a second solution, using equation 12, should be made. 
84. Short Cylinders, Flues, etc. When the cylinder or flue is 

1 / 
short, i.e., — <T 4 to 6, the effect of the heads should not be neg 

lected as in Carman's and Stewart's work, and Fairbairn's ap- 
proximate equation is applicable, or 

t 2 
w = 9fil5fiooj~ d (14) 

or transposing 



w I d f a 

t= '\—? — 7— (*5) 

>' 9,675,600 

If a cylinder under external pressure could be depended upon 
to fail only by actual crushing, instead of through collapse 

2 pt 
(buckling), then the formula w = — - — would apply, as in 

ijb 

internal pressure; remembering that under external pressure 
the stress p is compression. If this equation gives a lower work- 
ing pressure than (14) the flue designed by it will be safe against 
collapse. The rules of the Lloyd's Marine Register allow the 

1,075,200 t 2 
following pressure on boiler flues: w= j-j • This is 

Fairbairn's equation with a factor of safety of 9. The British 
Board of Trade rules allow a working stress in furnaces and flues 

2pt 
of about 4,000 when computed by the equation w = —7-. This 



222 MACHINE DESIGN 

is a little less than that allowed by the U. S. Board of Supervising 
Inspectors. 

Hence for the same allowable pressure under these two rules 

2pt 8,000/ 1,075, 200/ 2 

w = 



d d Id 

, or I = 134.4' i 

If, therefore, / < 134.4 t equations (1), (2), and (3) may be safely 
used. 

It will be observed that this relation limits the use of these 
equations to comparatively short flues. Thus a flue %" thick 
could only be 34" long to have these equations applicable. In 
practice long flues of large diameter are reinforced at short inter- 
vals by heavy rings of rolled or other section, known as collapse 
rings, thus making the flue consist virtually of a series of short 
flues, to which equations (1), (2), and (3) may be applied. 

Various Insurance and Government inspection departments 
give rules for proportioning flues and furnaces. These rules 
change from time to time, and if the boiler is to be insured in 
any company the specific rules prescribed by it should be consulted. 

Thus Lloyd's Register for 1906-7 gives 

1,075,200 t 2 

w _ when / >i2o/ 

I X a 

, 50(300/ — /) 

and w = - when / < 1 20 / 

a 

where /, / and d are all in inches. 

Various other authorities give similar equations with prac- 
tically the same coefficients. 

85. Corrugated Furnace Flues. Flues corrugated as in Fig. 
73 are very much stiffer against collapse than plain cylindrical 
flues, and with proper dimensions of corrugations may be safely 
made of any desired length. Their peculiar shape also permits 
of expansion and contraction under the influence of heat. When 
the corrugations are not less than 1 ]4 inches deep, and not more 
than 8 inches from centre to centre of corrugations, and plain 



TUBES, PIPES, CYLINDERS, FLUES, AND THIN PLATES 223 

portions at the ends do not exceed 9 inches, the U. S. B. S. I. 
allows a working pressure of 

14,000 / 

w = - 

a 

This is also the formula of the British Board of Trade. 

Lloyd's Register for 1907-8 gives a number of rules for design- 
ing various types of flues. 

The following references contain valuable practical informa- 
tion on this subject : 

Lloyd's Register of British and Foreign Shipping. 

" Steam Boilers," by Peabody and Miller. 

Rules and Regulations of U. S. Board of Supervising In- 
spectors. 

Rules and Regulations of the American Bureau of Shipping. 

Rules of the British Board of Trade. 

Rules of the Bureau Veritas. 

Seaton and Rounthwaite's Pocket Book. 

THICK CYLINDERS 

86. When the wall of a cylinder, which is subjected to 
internal or external fluid pressure, is thick relatively to the 
internal diameter, it can no longer be assumed that the stress 
in the wall is uniformly distributed over the cross-section, but 
it is greater at the inner surface and decreases to a minimum 
at the outer surface whether the pressure is internal or external. 
When the pressure is internal the stress is tensile, and when the 
pressure is external the stress is compressive. 

Many formulae have been deduced to express the relations 
between pressure, stress, and cylinder thickness. Of these, that 
of Lame, deduced in 1833, is perhaps best known. Clavarino's* 
modification of Lame's formula, which was published in 1880, is 
now much used and will be adopted in this work. 

Ordinarily the cylinder is subjected to either external or 
internal pressure alone; but in a gun tube, for example, which 

* The student is advised to read the discussion of thick cylinders given in Merri- 
man's "Mechanics of Materials," edition of 1906. 



224 MACHINE DESIGN 

has a hoop shrunk upon it, the more general case occurs in which 
the cylinder is subjected to both internal and external pressure. 

Let w 1 = the internal unit pressure. 
" w 2 = the external " " 
" r 1 = th.e internal radius of the cylinder. 
" r 2 = the external " ■ " " 

" /> 1 = the unit stress at the inner surface. 
" p 2 = the " " " outer " 
Then by Clavarino's equation the unit stress at any radius r is 

I r? w x - r 2 2 w 2 + ^ (w, - w 2 ) \ 
P = 7-1 2^ • ( l6 ) 

3(r%— r i) 

If the external pressure w 2 be zero, which is the most usual 
case, the greatest tensile stress is at the inner surface, and is 

a= tL^^^j (I7) 



or r , = 



r, L 3 A-4w,J • • • • ^ 



r, = 



Example. A cast-iron cylinder 20 inches in internal diameter. 
is to withstand an internal pressure of 1,000 lbs. per square inch. 
How thick must the wall be in order that the stress at the inner 
surface may not exceed 4,000 lbs. per square inch ? 

Here r =10, w x = 1,000 and p x — 4,000. Hence substituting 
in (18) 

= r r 3& + ^i -| h = IO f 3X4,000+1,000 -i|_ i2 g// 
1 L3 Pi —4 ^iJ L3 X 4,000 — 4 X 1, 000 J 

or the cylinder walls must be 2.8" thick. 

From (16) it is found that p 2 , the stress in the cylinder walls 
at the outer fibre, is 2,620 lbs. 

PRACTICAL CONSIDERATIONS 

87. Cast-iron pipes are widely used for underground water 
pipes and to some extent also for gas pipes, largely on account 
of their durability against corrosion. For steam, or for 
high pressures generally, cast-iron pipes are now seldom used 



TUBES, PIPES, CYLINDERS, FLUES, AND THIN PLATES 225 

because of their unreliability. For all ordinary purposes 
pipes made of wrought iron or steel are most used, although 
in special cases, such as marine work, copper and brass are 
preferred. 

Wrought-iron or steel pipes may be either lap-welded or butt- 
welded, the latter being commonly used for the smaller diameters, 
while steel piping may be " drawn" so that there is no seam, in 
which case it is known as "seamless drawn tubing." 

Standard Piping is designated by its nominal internal diameter. 
Thus standard i-inch gas pipe has a nominal internal diameter 
of 1 inch, and an external diameter of 1.3 15 inches. So-called 
standard wrought-iron piping may be used for pressures up to 
100 lbs. with safety. For still higher pressures, such as are found 
in high-class steam plants, thicker pipes, known as extra strong, 
are used. For hydraulic work, where pressures up to several 
thousand lbs. per square inch must be withstood, still thicker 
piping, known as double extra strong, is used. These heavy pipes 
are made by decreasing the internal diameter of the standard 
pipe, thus keeping the outside diameter and hence the screw 
threads for the flanges to one standard.* Thus an extra strong 
1 -inch pipe (nominal size) would have an internal diameter of 
.95 inches, and a double extra strong of the same nominal size 
would have an internal diameter of .587 inches, the external 
diameter remaining 1.3 15 inches in all cases. 

For large cylinders both for steam and hydraulic service, 
cast iron is still much used and probably will be for some time 
yet, on account of the ease with which complicated iron castings 
can be made and machined. In the case of steam-engine cylin- 
ders the thickness of the walls is fixed by considerations other 
than those of strength, such as stiffness and securing good cast- 
ings. The proportions of steam cylinders as fixed by practice 
are the best guide. An examination of current practice shows 
the average thickness of low-speed engines to be given by the 



* The student is referred to Kent's " Engineer's Pocket Book," or similar works, 
for full tables of standard sizes of pipes, flanges, etc. See also current trade 
catalogues. 
i = 



226 MACHINE DESIGN 

following, t = .05 d + .3 inch ,* where / = thickness and d = diameter 
in inches, when the steam pressure does not exceed 125 lbs. per 
sq. inch. 

Kent's " Mechanical Engineer's Pocket Book " gives the follow- 
ing as representing current practice, t=^.ooo/\dp-\-.^, where d = 
diameter in inches and p = pressure in pounds per square inch. 
If p be taken as 125 pounds this equation reduces to that given 
by Barr. 

Cast iron is also much used for the cylinders of hydraulic 
machines, although steel castings are better in general. In such 
cases equations (16) to (18) developed above, in common with all 
equations based on the theory of elasticity, should be used with 
caution when cast iron is selected for the cylinder. Further- 
more it must be borne in mind that the thicker the cylinder walls, 
the more liable are they to be porous in the interior, where made 
of castings. It is safer, therefore, as a rule, to carry a high work- 
ing stress, within safe limits, and insure sound castings, than to 
design thick walls which are open to suspicion, in order to get a 
theoretically lower stress. A 3-inch wall, for instance, with a 
working stress of 5,000 pounds per square inch is preferable to a 
4-inch wall with a working stress of 3,000 pounds per square inch. 
Care should also be exercised in cylinders made of castings to 
avoid excessive thickness of metal at any point, thus insuring 
sound castings. Thick castings of any metal are very liable to 
give trouble by leaking on account of porosity, if subjected to high 
pressures, and cast-iron cylinders are often lined with brass or 
bronze liners to obviate this difficulty. 

87.1. Pipe Couplings, Flanges, etc. Methods for securing 
the ends of pipes together have become of greater importance 
as higher steam pressures have been employed. The most usual 
method for accomplishing this purpose has been to thread the 
ends of the pipes (see Art. 57) and secure them together with 
either a cylindrical pipe coupling, a pipe union, or a pair of pipe 
flanges. All of these are in very common use. For pressures 
up to 100 pounds per square inch and pipes not over 12 inches in 

* See "Current Practice in Engine Proportions," by J. H. Barr; Transac- 
tions A. S. M. E., Vol. XVIII. 



TUBES, PIPES, CYLINDERS, FLUES, AND THIN PLATES 227 

diameter these may be used with success, but for higher pressures 
and larger diameters they are not so satisfactory. The union 
is used on small pipes only. 

In the ordinary screwed fitting of large size it is difficult to 
cut the thread accurately, and to screw the fitting on tight 
enough to prevent leakage at A, Fig. 73 (a). This can be rem- 
edied to some extent by making the threaded portion of the pipe 
long enough to project through the flange slightly, and then 
facing off pipe and flange so as to make a smooth surface, and 
permitting the packing or gasket (P) to cover up the screwed 
joint, as shown at B, Fig. 73 (a). Even this joint, however, is 
liable to leak if the workmanship is poor or if the flanges do not 
align properly. 

To obviate the difficulties of the screwed joint on pipe of 




Fig. 73 (a). 



Fig. 73 (b). 



Fig. 73 (c). 



larger diameter, the flanges are sometimes shrunk on as shown 
in Fig. 73 (b) (see also Art. 73). In order to insure tightness, 
and secure a firmer grip on the flange, the end of the pipe is 
usually expanded into the flange, as shown in Fig. 73 (b). The 
gasket usually covers up the joint between the pipe and the 
flange. This form of coupling is not well suited for high-pressure 
work, however, especially if the pipe is not machined on the out- 
side before the flange is shrunk on. When subjected to the 
heavy straining action incident to expansion and contraction, 
as in heavy steam mains, the pipe is sure to work in the flange, 
and leaking will ensue. These flanges are sometimes fitted with 
a recess,/? (Fig. 73 b), into which a strip of soft metal, such as 



228 MACHINE DESIGN 

copper, can be caulked to check small leaks; but this can hardly 
be considered satisfactory in high-grade work. A somewhat 
better grip of the flange is sometimes obtained by rolling the 
pipe into a groove in the flange, as shown at E, Fig. 73 (b) . 

In the so-called Van Stone joint (Fig. 73 c), the ends of 
the pipes themselves are flanged over and the joint made between 
the flanges so formed. The flanges F, F become clamps for 
holding the flanges proper together, and may be loose on the pipe. 
This last feature is a very useful one, as it greatly facilitates 
erection. This form of joint has been used with success in high- 
pressure work. 

For the highest grade of work, wrought-steel flanges are 
welded to the pipe, making the pipe and its flanges one piece. 
This construction, while expensive, is almost essential for large 
pipe and the highest pressures. 

To prevent the packing from blowing out, the flanges are 
sometimes fitted with a recess and tongue as shown at H, Fig. 73 
(b). This construction is essential for very high pressures, as 
in hydraulic work, but should be avoided if possible in steam 
lines, as it makes it difficult to renew the packing.* 

Although several efforts have been made to establish standard 
dimensions for pipe flanges, several systems are in common use 
in this country. The most important of these systems are the 
standards adopted by the Flange Standardization Committee, 
of the A. S. M. E., and that known as the Manufacturers' Standard. 
The first is used for pressures up to 125 pounds per square inch 
and the second for pressures up to 250 pounds per square inch. 
The student is referred to standard handbooks, and the catalogues 
of various manufacturers for details of these several systems. 
See also Transactions A. S. M. E., Vol. XXI. 

THIN PLATES 

88. General Theory. The theory of the stresses induced in 
thin plates, when subjected to load pressures, is one of the most 
uncertain portions of the mechanics of materials, due in part to 

* For a fuller description of Van Stone joints see articles by W. F. Fischer, 
Power, Feb. 23, 1909, and March 2, 1909. 



TUBES, PIPES, CYLINDERS, FLUES, AND THIN PLATES 229 

the complexity of the problem, and in part to the scarcity of 
corroborative experimental data. The subject has been investi- 
gated mathematically by Grashof, Bach, Unwin, Merriman, and 
others, and the experimental work of Bach, Benjamin, and others 
verifies in a measure some of their conclusions. The mathematical 
results obtained by various authorities differ mainly in the co- 
efficients, the general form of the equations being in most cases 
similar. Those due to Merriman* will be used in this treatise 
as they are simple, easy to apply, and give values as safe, generally, 
as any others. 

Let t = the thickness of the plate in inches; r = the radius 
of circular plate in inches; I = length and b = breadth of rect- 
angular plate in inches; p = maximum tensile stress; w = load 
per unit area in lbs. ; and P = concentrated load in lbs. Then 

For flat circular plates supported but not fixed at the edges, 
carrying a distributed load 



w 
t = r -V — for wrought iron or steel (1) 



9 w 
and t = r \ — — for cast iron ....... (2) 

^ 8 p v J 

For flat circular plates fixed at the edges (encastre), carrying 

a distributed load 



2 w 
t =■= r \j for wrought iron or steel .... (3) 



1 3 w 
and t = r \l for cast iron (4) 

^ 4 P 

For flat circular plates supported but not fixed at the edges, 
carrying a concentrated load P, which is applied to the centre 
of the plate over a small circle of radius r o so that P = tz r o 2 w o 



t — }\ \j i + 2 / g — — - for steel or wrought iron (5) 

I o — I I 

r~Q Q Y — 1 IV 

and / = r \M - + — log — — - for cast iron ... (6) 
> L8 4 e r J p 

-~~— . ___ . — . — ^ — . — , _ ^ — . — . — . — ■ « 

* See Merriman's " Mechanics of Materials," 1907 edition, page 409. 



230 MACHINE DESIGN 

For flat circular plates encastre carrying a concentrated load 

P, which is applied to the centre of the plate over a small circle 
of radius r. 

o 

I P-2 4 Y ~ 1 W 

t = r o *y \ — + —log e — — ° for steel or wrought iron (7) 

L-O O o — \ l 

and t = r^ - + - log e — -f for cast iron ... (8) 

In equations (5), (6), (7) and (8) w o is the pressure per unit 

P 
area on the small circle whose radius is r , i.e.. w n — — ,. The 

Tir 

value of w o should not exceed the elastic strength of the material. 
Example. A circular cast-iron plate 20" in diameter supports 
a load of 4,000 lbs. at its centre, the load being applied by a bolt 
whose head is 2" in diameter. How thick must the plate be, if 
simply supported, in order that the tensile strength in it shall not 
exceed 6,000 lbs. per square inch? 

Here P = 4,000; r = 10; r o = 1 and p = 6,000. 

P 4,000 _ 

• • w o = — ~2 = r-r— -2 = 1,265 lbs. 

n r n X 1 

r 10 

and lo g % — = / g e — = 2.3 whence from equation (6) 



ft 



9 -1 1,240 , 

+ -x 2.3 \-^- = 1.14 
4 ^ j 6,000 



Rectangular Plates. If 2/ and 26 are the length and breadth 
of a rectangular plate, then for plates supported but not fixed, 
and for a uniformly distributed load w per square inch 



i -\ lh \ n2 , W t,2w . . (9) 



w 

v +> 

and for fixed edges 



= lb^ 



3 w 



2(P+b')p 



(10) 



TUBES, PIPES, CYLINDERS, FLUES, AND THIN PLATES 231 

If / = b equation (9) reduces to 

w 



-'VI 



and equation (10) reduces to 

'-'V*? <») 

The above equations, as before stated, are not to be relied on 
implicitly, but will serve as approximate guides only. This is 
particularly true in cast materials where heavy ribbing is used 
and where trained judgment is perhaps the best guide. 

89. Flat Stayed Surfaces. One of the most important cases 
of flat plates occurs in boiler work, where large flat areas are held 
against pressure by stays at regular intervals over the surface. 
These stays are usually screwed into the plate and the projecting 
end is slightly riveted over to insure steam tightness. The various 
Inspection Bureaus and Insurance Companies give practical 
formulae for the design of such plates, and these can be safely 
used. Thus the U. S. Board of Supervising Inspectors* and the 
American Boiler Makers' Association rules give for steel plates 

cxf 
w = -7- (13) 

Where w = pressure in lbs. per sq. in., t = thickness of plate in 
sixteenths of an inch, 5 = greatest pitch of stays in inches, and C = 
a constant as below given 

C = 1 1 2 for plates yq " thick and under. 
C = i2o " " over ^" thick. 

C = i4o " with stays having a nut inside and outside. 

C = i6o " " " " " washers .5 as thick as 

the plate and of a diameter at least .5 the greatest pitch. 

* See General Rules and Regulations of U. S. Supervising Inspectors; also 
Rules of American Bureau of Shipping. 



CHAPTER X 
CONSTRAINING SURFACES 

90. General Considerations. As the various members of a 
machine must move with definite relative motion, they must be 
retained in correct position by constraining surfaces. Thus 
a shaft is held in position by bearings which locate its axis 
of rotation, and by collars which prevent motion endwise. The 
relative motion of a pair of constrained members may be that of 
sliding, as in the case of an engine crosshead and its guide; 
rotation, as in the case of a shaft journal and its bearing; rolling, 
as in roller and ball bearings; or a combination of some of these 
as illustrated in certain forms of cams, where both sliding and 
rolling exist. Dry metallic surfaces, under any appreciable load, 
even when smoothly machined, will not slide over each other 
without abrasion. It is therefore necessary to keep rubbing 
surfaces separated by a thin film of some kind of lubricant, and 
the whole subject of the design of constraining surfaces is closely 
connected with the theory of lubrication.* 

It has been pointed out in Chapter IV, that when bath or 
forced lubrication is maintained, the friction between two rubbing 
surfaces is independent of the character of the material of which 
the surfaces are composed; but when the surfaces are " imper- 
fectly" lubricated the frictional resistance depends somewhat on 
the metals used. Experience has shown that like metals usually do 
not rub together well. Thus steel on steel (except when hardened) , 
steel on wrought iron, or cast iron on cast iron, are poor combina- 
tions except where the velocity is low and the pressure light. If two 
rubbing surfaces of cast iron can be run together for some time 
without cutting, they take on hard glazed surfaces which will run 
well together. This is well illustrated in slide valves and pistons of 

* See Chapter IV. 

232 



CONSTRAINING SURFACES 233 

steam engines. Care must be exercised that the surfaces are well 
lubricated when first put in service. Soft steel and wrought iron 
will both run well on hardened steel, and hardened steel may 
be run on hardened steel at very high pressures and velocities, if 
the surfaces are ground true, and polished. Steel and wrought 
iron will run very well on brass or bronze. The alloys of copper, 
tin, zinc, antimony, lead, etc., commonly known as anti -friction 
or babbitt metals, run extremely well with steel or wr ought- 
iron journals.* Innumerable alloys of this kind are upon 
the market under different names. They can be made of any 
degree of hardness, depending largely upon the proportion of 
antimony used. Very hard alloys of this kind are sometimes 
known as white brass. In using babbitt metal for heavy pres- 
sures, care should be exercised that the particular alloy selected 
is hard enough so as not to flow under the applied pressure. 
Other materials, such as wood, are sometimes used for rubbing 
surfaces. The conditions which influence the selection of 
materials for rubbing surfaces, and the practical considerations 
governing their application, will be more fully discussed in con- 
nection with the several forms of constraining surfaces. 

The most corrlmon forms of motion in machines are rectilinear 
translation and rotation; therefore the most important forms of 
constraining surfaces are 

(a) Sliding surfaces, for the constrainment of rectilinear 

motion. 

(b) Journals and bearings, for the constrainment of motion 

of rotation. 

SLIDING SURFACES 

91. Forms of Sliding Pairs. The stationary member of a pair 
of surfaces, which have relative sliding motion, is usually called 
the guide, while the moving part has various names depending 
on the service, as the ram of a shaping machine, the table of a 
planing machine, or the crosshead of an engine. The general 
term sliding member will be used here to denote the moving 



* See Kent's "Mechanical Engineer's Pocket Book" for detailed analysis and 
properties of some of the best known alloys. 



234 MACHINE DESIGN 

member. Sliding pairs may be classified by the degree of lateral 
constrainment afforded the slider by the guides, and this may be 

(a) Partial lateral constrainment. 

(b) Complete lateral constrainment. 

In either case the rubbing surfaces of the guide and sliding 
member may be either square, angular, or circular. Thus Fig. 
74 (a) shows a form of angular guide much used on planing ma- 
chines while Fig. 74 (b) shows a set of square guides for a similar 
purpose. In each case the lateral constrainment is only partial, 
the tendency of the platen to raise being resisted by gravity. 
Fig. 75 (a) shows the crosshead of a steam engine with an angular 
guide. Here, lateral constrainment is complete. Fig. 75 (b) is 
also a steam-engine crosshead with circular guiding surfaces. 
This form of surface may be considered as a special form of the 





Fig. 74 (a). Fig. 74 (b). 

angular type. If the circular guiding surfaces have a common 
centre at O, the crosshead is prevented from rotating around O 
only by the connecting-rod ; and as long as it is so held from 
rotating the lateral constrainment is complete. If the surfaces 
have different centres as O t 2 , it is obvious that rotation cannot 
take place. Figs. 76 (a) and 76 (b) show square and angular 
guides where constrainment is complete. 

The characteristic which distinguishes the square guide from 
the angular one is that in the square guide two sets of adjustments 
must be made to compensate for wear, while in the case of the 
angular guide one set only is needed. Thus in Fig. 76 (a), vertical 
wear must be compensated for by lowering the piece A, while 
lateral wear is taken up by the set screws C which press against 
the wearing strip or gib B. In Fig. 76 (b) lateral and vertical wear 
are both compensated for by the set screws C which press upon 
the gib D. Sometimes D is made tapering and provided with a 



CONSTRAINING SURFACES 



2 35 



screw adjustment so that it can be moved endwise, thus com- 
pensating for wear. In such cases the set screws C are omitted. 
As to the relative merits of square and angular guiding sur- 
faces, it may be said, in general, that square surfaces are easier to 
machine and fit than the angular ones. There are many places, 
however, such as the cross slides of lathe carriages, where the 
angular guide is much more convenient. In places such as lathe 
beds the V guides commonly used have the advantage of auto- 
matically taking up lost motion, no matter how badly they are 
worn. But, as a rule, the bearing surfaces of such V guides are 
very small and wear soon begins to be apparent, especially as the 
wear from the carriage is usually concentrated on a short por- 




Fig. 75 




Fig. 76. 



tion of the bed. There is a tendency among manufacturers to 
discard the V guide in favor of flat surfaces. English practice, 
especially in large tools, is in advance of American practice in this 
particular. A combination of V and flat guides is also often 
used. 

92. General Principles. If a short block, Fig. 77, slides 
backward and forward upon another member B, carrying a fixed 
load P, it is evident that, if the material in A and B were homo- 
geneous and the velocity were uniform throughout the stroke, 
the frictional resistance and consequent wear would be practically 
uniform over the whole surface of B. These conditions are 
difficult to attain and seldom occur in practice. Since A must 
be stopped and started at each end of the stroke, it follows that 
the velocity cannot be uniform; although in some machines such 
as plate planers this condition is approximated. Usually, how- 



236 MACHINE DESIGN 

ever, the velocity varies from zero at the beginning to a maximum 
somewhere near the middle of the stroke, as in the case of engine 
crossheads, shaping machines, etc. Again, the load P may vary 
greatly. Thus in the steam engine, the^ normal pressure P be- 
tween the crosshead and its guide is zero at each end of the stroke 
and a maximum near mid-stroke. The velocity of the crosshead 
also varies from zero at each end of the stroke to a maximum 
near mid-stroke. In the ordinary case the greatest frictional 
resistance* and wear will occur near mid-stroke, because both 
velocity and normal pressure between the bearing surfaces ■ are 
greatest at this position. If the crosshead could be made the 
same length as the guide, the unit bearing pressure, at the middle 
of the stroke, would be practically uniform over the whole surface, 
and would be small compared to the unit normal pressure attained 
when the crosshead is short. For positions of the crosshead near 
mid-stroke the wear would be approximately equal over the whole 
surface, and much less than when the crosshead is very short, 
but still theoretically greater than at the end positions, when both 
velocity and normal pressure are zero. It has been found by 
experience that when the sliding block and guide are made 
the same length, the wear, even under varying load and ve- 
locity, is very small, and more uniform over the entire contact 
surfaces. 

It is seldom possible, however, to make the sliding member 
the same length as the guide. Thus in lathe carriages, the rams 
of shaping machines, and the tables of planing machines, the 
sliding member is, in some machines, shorter than the guide, and 
in other machines longer. In most cases of this kind the wear 
is liable to be greater on one part of the guide, or sliding member, 
than on another. Thus in the case of a shaping machine the 
ram seldom operates at full stroke, and the wear on the back end 
of the ram is very small, the result being that when appreciable 
wear takes place on the forward end of the ram, and the guides 
are readjusted to compensate for the same, the back end of the 
ram will not pass through the guides at all, hence the 

* See Art. 32. 



CONSTRAINING SURFACES 



237 



adjustment must be somewhat slack, and accurate work 
cannot be done. • In other machines the excessive local wear 
comes on the guide, and a similar result occurs. Professor Sweet * 
has corrected this difficulty, in certain machines which he has 
built, by reducing the wearing surface on that portion of the slid- 
ing member or guide, as the case may be, where the tendency to 
wear is least. He has suggested the following convenient method 
of laying out the wearing strips on the surface of a sliding member. 
Fig. 78 shows a sliding surface such as is found on the ram of a 
shaping machine, where little wear occurs on the back or right- 
hand end as here shown. The shaded portions represent the 
parts of the surface which have been relieved, leaving the wearing 

' X 



Fig. 77. 




a 9 

o .2 

o o 

2 a 



(a) 




Fig. 78. 



Fig. 79. 



strips S, S ± and S 2 , etc. To lay off the surface, draw the diagonal 
a, b across the surface to be relieved. From a draw the line ac, 
making any convenient angle with the horizontal. Lay off ce 
equal to the width of the face x. Draw de parallel to ac and take 
the vertical distance above the point of intersection of ab and de 
for the first gap, and the corresponding vertical distance below 
the point of intersection for the first wearing strip, repeating this 
operation to the end of the surface. Similar wearing strips should 
be cut in the opposite direction on the other member, if it is 
comparatively long; but where a short block slides in a long 
guide, the guide only need be relieved. 

93. Bearing Pressures on Sliding Surfaces. It is pointed out 
in Article 32 that the tendency of a loaded flat surface to expel 
the lubricant is resisted to a certain degree by the viscosity of 



* Professor Sweet has embodied some of his experience in this line in a little 
book called "Things that are Usually Wrong," which will well repay reading. 



238 MACHINE DESIGN 

the lubricant, and its power to adhere to the stationary member. 
This resisting power is much less marked in sliding surfaces than 
in rotating surfaces, as here the motion is intermittent. It is 
difficult therefore to lubricate sliding surfaces as efficiently as 
rotating surfaces, and, in general, they must be considered as 
" imperfectly " lubricated surfaces. The unit bearing pressure 
that can be sustained by sliding surfaces is, therefore, much less 
than can be borne by rotating journals. Further, it is difficult to 
obtain initially true sliding surfaces and, as pointed out above, 
very difficult to maintain their accuracy under service. The 
sliding part, and also the guides themselves, should, therefore, be 
designed for rigidity;' in fact considerations of strength seldom 
need to be considered, but the guides should be so stiff that 
localized pressure will not occur. It is not surprising, in view 
of these considerations, that the allowable bearing pressures as 
fixed by practice vary greatly, even with similar classes of work. 
Owing to the difficulties of lubrication and compensation for wear, 
it may be stated, as a general principle, that the bearing pressure 
must be kept so low that wear is inappreciable, if accurate sur- 
faces are to be maintained. 

The following are average values of bearing pressures for 
different forms of sliding surfaces, as fixed by practice: 

Crossheads,* stationary slow-speed engines 30 lbs. to 50 lbs. 

high " " 10 " " 30 " 

" marine engines 50 " " 75 " 

94. Lubrication of Sliding Surfaces. Sliding surfaces are very 
difficult to lubricate efficiently on account of the " wiping" action 
of the sliding member. In high-speed engines, bath lubrication 
is commonly obtained by enclosing the running parts, and allow- 
ing them to run in what practically amounts to an oil bath. 

Where this cannot be done, care must be exercised in the 
manner in which the lubricant is supplied. If possible, when 
the guide is horizontal, the lubricant should be supplied near 
the middle of the guide. The oil grooves in the moving part should 
also be given careful consideration. From the theory of lubrica- 

* See Trans. A. S. M. E., Vol. XVIII, page 753. 



CONSTRAINING SURFACES 239 

tion it is evident that the oil channels on all constraining surfaces 
should be at right angles to the direction of motion, wherever the 
velocity is great enough to draw lubricant between the surfaces. 
If made otherwise their effect is to relieve any tendency to form 
a pressure film. The grooves in crossheads, and other sliding 
members, should, therefore, be made as in Fig. 79 (a) and not as 
in 79 (b). In either case the grooves should be stopped some 
distance from the edge of the surface so as not to facilitate the 
escape of the oil. When the load is so heavy that forced lubrica- 
tion must be used, the system of grooves shown in Fig. 79 (b) is 
correct; the oil being forced in at O. Care should also be taken 
that the outer edges of the slider, and the edges of the oil grooves, 
are chamfered so as to assist the entrance of the lubricant. If 
the edges are square and sharp their scraping effect may seriously 
impair the lubrication. Where the guiding surfaces are very 
long, as in planing machines, oiling devices such as rollers dipping 
in an oil pocket, placed at intervals along the guides, are very 
effective. 

JOURNALS AND BEARINGS 

BEARINGS 

95. Forms of Bearings. The part of a machine frame, or 
other member, which constrains a rotating member, such 
as a shaft, is known as a bearing. That portion of the 
rotating member which engages with the bearing is known as a 
journal. Journals are necessarily circular in all cross-sections, 
but their profile may be cylindrical, conical, spherical, or even 
more complex in form, as in the case of thrust bearings. (See 
Art. 105.) 

One or more of the following considerations affect the design 
of the bearing proper: — 

(a) Rigidity, in order that the alignment may not be seriously 

affected by deflection. 

(b) Strength, to resist rupture under the greatest loads. 

(c) Adjustment, to compensate for wear. 

(d) Formation and maintenance of an oil film. 

(e) Automatic adjustment, to insure alignment. 



240 MACHINE DESIGN 

(a and b) . The inside diameter, or bore of the bearing, and also 
its length are fixed by the dimensions of the journal which engages 
with it; and the required strength and rigidity may be secured 
by a proper distribution of metal in accordance with the general 
principles discussed in Chapter III, and which apply to all forms 
of bearings, as far as strength and stiffness are concerned. 

Usually the question of strength does not enter into the design 
of the main part of the bearing. If, however, the cap A, Fig. 80, 
should be called upon to carry the load, as is often the case, its di- 
mensions should, in general, be checked for strength, and its design 
should be such that stiffness is secured. The exact distribution of 
the pressure over a bearing is not known;* but the assumption 
that the cap is a beam loaded at the centre and of a length equal 
to the distance between the cap bolts will give dimensions on the 
safe side for strength and deflection. The greatest bending 
moment and deflection for such beams are given in Case IX, 
Table 1. It is impossible to adjust the cap bolts so as to be sure 
that the load is uniformly distributed among them, and the un- 
certainty of the initial stress due to screwing up the nuts makes 
the problem more difficult. For this reason the cap bolts should 
be designed to carry more than the apparent load. If only two 
bolts are used each should be designed for two-thirds of the total 
load; if four are used each should be able to carry one-third of 
the load with an apparent stress of not more than 6,000 lbs. per 
square inch. 

The last three items, c, d and e, affect the form of the bearing. 
Consider first c and d. It is evident that the metal of the bearing 
will wear away most rapidly in the line of greatest pressure, hence 
adjustment for wear should also be along this line. It follows 
also that the bearing should be parted at right angles to the line 
of greatest pressure. Thus, if the load on the shaft be a simple 
vertical load P, as in Fig. 80, wear will take place only on the 
bottom half of the bearing. If this wear is so small as not to 
interfere with the alignment of the shaft, or if all the bearings 
on the shaft wear uniformly, adjustment may be made by lowering 
the cap A. If the shaft must occupy a fixed position relative to 
the frame of the machine, alignment must be maintained by 

' * See Article 27. 



CONSTRAINING SURFACES 



241 



raising the lower bearing surface. Where this is desirable the 
lower wearing surface is usually made separate from the pillow- 
block, as in Fig. 82, thus allowing the bearing to remain fixed 
in position, while the wearing part may be raised to compensate 
for the wear. If the load P, Fig. 80, be in an upward direction, 
all necessary adjustment may be made by means of the cap. 

It was shown in Articles 32 and 33 that a journal will auto- 
matically tend to form a film of lubricant between itself and the 
bearing. If the conditions under which the lubricant is supplied 
are correct, fluid pressure may thus be created between the 
journal and bearing provided the surface of the bearing is con- 
tinuous for some distance on each side of the line of action of the 




Fig. 80. 




load. The greatest pressure will be found near this line of action. 
It is evident that the bearing shown in Fig. 80 fulfills both these 
requirements for vertical load either upward or downward; but 
is unsuited for lateral pressure from the standpoints both of ad- 
justment for wear and lubrication. 

Suppose, however, that the journal carries a heavy vertical 
load P (Fig. 81), and is subjected at the same time to a heavy 
horizontal belt pull P v The resultant of these forces is P 2 , and 
the arrangement of parts shown in Fig. 81 is correct for motion 
of rotation in either direction. If P x be reversed in direction 
the resultant of P x and P 2 will be P 3 , and the arrangement is not 
correct for adjustment against wear, and very defective as far as 
lubrication is concerned, as the surface is broken near the point 
16 



242 



MACHINE DESIGN 



where the greatest film pressure should exist. Bearings of this 
form are often used in steam-engine work, and in such cases 
the force P x due to the steam pressure on the piston, is 
continually reversed in direction. Another adjustment for 
a similar case is shown in Fig. 82. Here the shoe or bottom 
" brass" can be raised up by introducing thin "shims," or liners, 
underneath it; while lateral wear can be taken up by setting out 
the " cheek pieces" B, by means of the wedges D. Provision is 
thus made, by this arrangement, for taking up wear in all direc- 
tions and keeping the shaft accurately aligned and located. For 
horizontal pressures in either direction the resultant P 3 passes 
close to the point at which the bearing is parted; and hence the 
best conditions for lubrication do not exist. Pressure films more 




Fig. 83. 



or less perfect, depending on the oil supply, will form on the lower 
shoe, but the continual reversing of the lateral pressure P, hardly 
allows time for the formation of pressure films on the cheeks. 
These reversals in pressure, however, allow the lubricant to be 
carried by the shaft, first under one cheek, and then under the 
other, thus lubricating them effectively. 

Sometimes a bearing consists of a conical bushing split at 
some convenient place, as shown in Fig. 83. By releasing the 
nut A y and screwing up on B, the bushing may be forced into 
the frame C, thus closing the bore of the bushing slightly and 
compensating for wear. It is obvious that once the bore of the 
bushing is worn eccentric, no amount of taking up can rectify 
its shape; in fact taking up wear in this manner tends to destroy 
the fit of the journal in the bearing. Occasionally the journal 



CONSTRAINING SURFACES 



243 



itself is made conical, and adjustment for wear is made by moving 
the shaft endwise. The application of such bearings is limited 
to short shafts, such as machine-tool spindles. 

Machine bearings are made in many forms, depending on 
the location and service. The bearings are sometimes split 
into three pieces, and various other means of compensating for 
wear are used, but the fundamental principles outlined above 
regarding the point where the bearing should be parted apply 
to all forms. 

Consider the last item (e, automatic adjustment). In long 
lines of shafting, which tend rapidly to get out of alignment, it 
is desirable that the bearing be so constructed as to adjust itself 
automatically to the chang- 
ing position of the shaft, 
in order to avoid localized 
pressure, which would re- 
sult in heating. In fast- 
running machinery, also, 
such as countershafts, dy- 
namos, and motors, where 
perfect alignment is neces- 
sary, self-adjusting bearings 
have been found almost 
essential. Fig. 84 shows a 
bearing of this kind as used in dynamo and motor bearings. 
The sleeve A has a spherical surface turned upon the outside, 
the centre of the surface being at O. This surface engages with 
a similar surface bored in the outer casing B. The sleeve may 
swivel in any direction, but the centre line of the shaft must 
always pass through O. When a shaft has only two bearings of 
this kind it is evident that perfect alignment can be secured, 
within the range of motion of the sleeves. Similar devices are 
used in the case of long shafting, where many bearings must be 
used. It is obvious that the fundamental principles regarding 
adjustment for wear and maintenance of the oil film, apply to all 
bearings of this form also. 

96. Practical Construction of Bearings. It was shown in 




Fig. 84. 



244 MACHINE DESIGN 

Article 88 that metal such as brass, bronze, and the white alloys 
make excellent bearing surfaces for wrought-iron or steel journals, 
on account of their anti-friction qualities. It is to be noted that 
even in the case of perfect lubrication, where the character of the 
rubbing surfaces is less important once the oil film is established, 
care must be exercised in the selection of the material for the 
bearing surface, in order that abrasion may not occur before the 
film is formed, or in case of failure of the film. There is a further 
advantage in having the bearing surface softer than the journal, 
in that it is very desirable to have the journal maintain its form 
against wear, which it is more likely to do when rubbing against 
a soft surface than it would against one harder than itself. The 
bearing itself should be rigid, so as to insure proper alignment of 
the shaft. Rigidity, against even moderate pressure, could not 
ordinarily be attained if the entire bearing member were made 
of the white alloys, and economy prohibits the use of brass and 
bronze for the entire bearing. It is customary, therefore, to 
make the main body of the bearing of cast iron (or sometimes a 
steel casting), and to fit into it wearing surfaces of the softer 
metals. These wearing surfaces may be either rigidly attached 
to the main castings or may be removable. In Fig. 80 is shown a 
bearing of the type commonly used for heavy shafts when the 
babbitt-metal lining is rigidly attached by means of dovetail 
shaped recesses, into which the babbitt is poured in a molten 
state. The necessary shrinkage due to cooling, which would 
leave the lining loose in the recesses, is usually overcome by 
hammering the babbitt, when cold, till it again fills the recesses, 
and then boring the babbitt to size. For cheap work the lining 
is often cast to size on a metal mandrel and no further work put 
upon it, but for all good work the bore of the lining is cast small 
enough to allow of hammering or peening, and then boring to a 
smooth surface. Fig. 81 shows removable linings of brass or 
bronze which are circular in section, and are prevented from 
turning when in place by the parting piece B. This parting 
piece, or "liner," also permits taking up wear by reducing 
its thickness as occasion requires. Fig. 82 shows an ar- 
rangement of wearing surfaces common on horizontal steam- 



CONSTRAINING SURFACES 



245 



engine bearings. The cap C is babbitted with some form of 
cheap metal since there is no wear upon it, all the pressure being 
either downward or side wise. The " quarter boxes" B, and the 
lower box or shoe A, may be of brass or bronze, or of cast iron 
lined with babbitt. Where there is danger of the boxes breaking, 
through pounding by the shaft, and where it is desired to use a 
babbitt metal, they may be made of brass or bronze and babbitt- 
lined. When cast-iron wearing surfaces are used, and compensa- 
tion for wear is important, as in the case of machine tools, it is 
customary to make the wearing surfaces removable as indicated 
in Fig. 81. For less accurate work the bearing surface is part of 
the main casting itself, machined to the required size Hardened 
steel bearing surfaces are obtained by making circular shells or 
"bushings," of the required internal diameter, and of sufficient 
thickness to insure strength. These bushings are forced into 
openings in the main casting and no provision for taking up wear 
is made. If the forcing operation closes the bore of the bushing, 
it is "lapped" out with emery and oil to the required size. Where 
the bearing must work under water, as in the case of a propeller 
shaft or the lower bearing of a vertical turbine water wheel, a 
lining of lignum vitae or other hard wood is often used. The 
surrounding water furnishes the only lubricant necessary in such 
cases. A detailed description of the many arrangements of 
bearing surfaces is beyond the scope of this treatise. 

When the bearing must work under trying conditions, as on 
shipboard or in a heated room, and there is some question as to 
whether the heat of friction will be dissipated by radiation, the 
bearing is cast hollow so that water may be circulated around it, 
thus carrying off the heat and maintaining the lubrication. In 
an emergency, water may be allowed to run over the outside 
of the bearing, accomplishing the same purpose. High-grade 
marine work, and large stationary-engine installations, are often 
equipped with a complete system of water circulation on the most 
important bearings. 

JOURNALS 

97. Theoretical Design of Journals. The considerations 
affecting the design of any journal are one or more of the following: 



246 MACHINE DESIGN 

(a) Strength to resist rupture. 

(b) Rigidity, or stiffness, to prevent undue yielding. 

(c) Maintenance of form against wear. 

(d) Maintenance of lubrication. 

(e) Radiation of the heat due to frictional resistance. 

The first two considerations, strength and rigidity, are 
covered by the general principles laid down in Chapter III, and 
are more fully considered in Chapter XI, where the special prob- 
lems in connection with shafts are discussed. Economy of 
material dictates that the minimum diameter of shaft be consistent 
with the applied bending and twisting moments. 

The third consideration (c) particularly affects such journals 
as those on the spindles of grinding machines and machine tools 
generally, where the accuracy of the product depends on the accu- 
racy of the journals. Usually, in such cases, the wearing surface 
must be so great, in order to reduce the wear to an inappreciable 
amount, that the consideration of strength does not enter into the 
computations. 

The considerations, (d) and (e), are closely correlated. It 
was shown in Articles 32 and 3$ that if the unit bearing pressure 
on the journal is not too great, the lubricant, because of its 
viscosity, may be drawn in between the journal and the bearing, 
thereby reducing the frictional resistance. This frictional re- 
sistance can never be reduced to zero even with perfect lubrica- 
tion. The energy thus absorbed appears as heat, and is radiated 
to the surrounding air by the metallic surfaces of the bearing, 
the temperature of which rises till the rate of radiation equals 
that at which heat is being generated. In well designed ma- 
chinery the temperature of the bearing should not exceed 150 F. 
The raising of the temperature of the bearing has a tendency to 
lower the viscosity of the lubricant, and if the bearing becomes 
too hot, the lubricant becomes so thin that the pressure squeezes 
it out completely, and failure of the bearing by abrasion occurs. 
It is evident, therefore, that a journal of given dimensions may 
carry a given load very satisfactorily under certain conditions, and 
fail absolutely under others, the same lubricant being used in 
each case. The consideration of the proper radiation of the heat 



CONSTRAINING SURFACES 



247 



generated is, therefore, most important. It may be assumed, without 
serious error, that the rate of radiation of heat is proportional to 
the projected area of the bearing. The number of heat units 
which will be radiated from a unit of surface, at any given 
difference in temperature between the bearing and the surround- 
ing air, is a fixed quantity for any set of conditions; and if the 
heat of friction per unit area is greater than can be radiated 
at the desired bearing temperature, the temperature of the bearing 
must rise till equilibrium is obtained. It follows therefore that 
for any desired bearing temperature the work of friction per unit 
of projected area of bearing must not exceed the rate of radiation 
per unit of projected area, or 

zr 

fi w V = K or w V = — (1) 

H 

where y. is the coefficient of friction, w the load in pounds per 
unit of projected area, V the velocity of rubbing in feet per 
minute, and K the rate of radiation per unit of projected area in 
foot pounds per minute, to be determined experimentally. 

It is to be especially noted that if y be considered as constant, 
increasing the diameter of a journal (the number of revolutions 
and the total load remaining constant) does not materially affect 
the development or dissipation of heat, since the velocity of rub- 
bing is increased in the same ratio as radiating surface is in- 
creased. If, however, the bearing be lengthened, the radiating 
surface is increased and the work of friction remains unchanged, 
with the same total load as before. This last statement, while 
true for imperfectly lubricated surfaces, is only approximately 
true for bearings with perfect lubrication as will be seen 
presently. 

The amount of heat which will be radiated from a bearing 
has been experimentally determined by Lasche.* The curves 
shown in Fig. 84 (a) are those shown in his Fig, 57, transformed 
into English units, and with the scale of radiation further modified 
so as to read in foot pounds per square inch of projected area 
per second, instead of per square inch of actual bearing surface. 

* See Traction and Transmission, January, 1903, page 52. 



248 



MACHINE DESIGN 



Curve i represents actual experimental results, with bearings of 
the usual proportions, in still air. Curve 2 is for bearings which 
are connected to large iron masses, or which are ventilated by air 
currents. Curve 3 was calculated from theoretical considera- 









































200 










































/ 
































180 






f 




































































160 








































fie/ 














w 


















140 




5 














33 

&4- 




















1 
09/ 












V / 






















120 


is 

£1 


f | 












V 
































4 










& 




























y 






*$ 




















100 
















<\& 


9y 




































*/ 






















80 






































GO 










































































40 
















































































































20 






















• 























































2 

60 



4 6 8 10 12 14 16 

= RADIATION IN FT. LBS. PER SEC PER SQ. IN. PRO J. AREA. 

Fig. 84 (a). 



tions. It gives the radiation from a very thin bearing or sleeve 
and indicates that radiation is more effective as the bearing be- 
comes thicker, as might be expected; for metal is a better con- 
ductor of heat than air, and hence the thick bearing more easily 



CONSTRAINING SURFACES 



249 



carries the heat away to a greater radiating surface. The values 
obtained from these curves may therefore be used for K in 
equation (1). Lasche points out that though these experiments 
represent only a limited variety of conditions, they are probably 
on the safe side and will serve at least as a very useful check in 
designing. 

If, in designing a journal, the value of n can be deter- 
mined, equation (1) and Fig. 84 (a) give the relations which 
must exist between the velocity and pressure in order that the 
safe bearing temperature may not be exceeded; or if the pressure 
and velocity are fixed by other circumstances, Fig. 84 (a) indicates 
whether radiation must be assisted by artificial means, such as 
water circulation or currents of air. 

98. Imperfectly Lubricated Journals. It has been shown in 
Articles 32 and 33 that the value of /*, for imperfectly lubricated 
surfaces, is a very variable quantity, even for the same simulta- 
neous values of velocity and pressure. Not only does it vary with 
velocity, pressure, and temperature, but the regularity of the 
oil supply (over which the designer has little control) affects it 
much more seriously. Further, bearings running under the 
same nominal load and velocity give widely different values of 
frictional resistance and temperature rise, depending on whether 
the load is constant or intermittent, or whether the motion is 
steady or vibratory, etc. Notwithstanding this, equation (1) may 
be made to serve as a useful check in doubtful cases by assum- 
ing a safe value of /-/-. 

The assumption is sometimes made that ;j- is a constant ; and 

formulae of the form w V = — = C, where C is a constant that 

has been determined from practice, are much used. Thus if w be 
expressed in pounds per square inch of projected area and V in 
feet per minute, Mr. Fred W. Taylor* gives for mill work C = 
24,000; and says that C = 12,000 is not safe for cast-iron bearings 
with ordinary lubrication. If the rise of temperature in the bearing 
be taken as 75 and P- be taken as .015, which is ordinarily a safe 

*»— ^— — — h i- - i' ■■ ■ ■ ■ ■ ■ ' -~ ■ ■ ■ — ■ ■■ . — ... .. '■■■'■ ■'■ 

* Transactions A. S. M. E., Vol. XXVII. 



250 MACHINE DESIGN 

value, then from curve 1, Fig. 84 (a), K = 222, whence C = — = 

A* 
222 

= iq.ooo. From curve 2, K = 384 whence for ventilated 

.015 D ° 

384 
bearings C = = 25,600. These values agree with Mr. 

Taylor's limits better than would be expected. 

All formulae of this empirical form must be considered, as 
far as imperfectly lubricated journals are concerned, as applying 
only to the conditions and range for which they have been found 
true, and for which />. is apparently constant. This is more 
evident when the wide variation of the value of such constants 
as determined by practice is considered. Thus Mr. H. G. Reist 
gives, as the practice of the General Electric Company on generator 
bearings, a limiting value of C = 50,000 for bearing pressures from 
30 to 80 pounds per square inch. Mr. H. P. Been gives the 
practice of one of the largest Corliss engine builders as C = 
60,000 to 78,000 for bearing pressures not higher than 140 
pounds per square inch. 

Unwin, page 249, gives values of a similar constant, /?, which 
corresponds to the following values of C: 

Locomotive Crank Pins 250,000 to 375,000 

Locomotive Axles 200,00c 

Marine Engine Crank Pins 50,000 to 75,000 

Stationary Engine Crank Pins i5>ooo to 50,000 

Railway Carriage Axles 75>ooo to 100,000 

Crank Shaft Bearings 7>5°° to 20,000 

The great variation in these values of C is no more than might 
be expected in view of the foregoing, and also in view of the 
difference in lubrication and in radiating capacities of bearings, 
due to material, form, and location. While, therefore, these 
coefficients may form a guide, and while doubtful cases may be 
checked for heating by equation (1), care should be exercised that 
the bearing pressure is kept within the limits which will admit 
of good lubrication. The allowable bearing pressures as fixed 
by practice for various classes of machines are given in the follow- 
ing table, and it may be noted that these are more accurately 



CONSTRAINING SURFACES 



251 



known than the values of /*, or the values of the coefficient of 
radiation K. 

Economy in the use of material and the importance of mini- 
mizing the work of friction suggest that the diameter of the 
journal shall be as small as is consistent with strength and stiff- 
ness. With the diameter of the journal determined by these 

TABLE XIII 

BEARING PRESSURES FOR VARIOUS CLASSES OF BEARINGS 



Class of Bearing and Condition of Operation. 



Bearings for very slow speed as in turntables in 
bridge work 

Bearings for slow speed and intermittent load as in 
punch presses 

Locomotive Wrist Pins 

Locomotive Crank Pins 

Locomotive Driving Journals 

Railway Car Axles 

■»t • -c> • T\>r • -d • ( Naval Practice. . 
Marine Engine Mam Bearings { Merchant Practice 

Marine Engine Crank Pins 

f (high speed) 
Stationary Engine Main Bearings \ for dead load.* 

[ for steam load . 

((high speed) 
overhung crank. . 
centre crank 

Stationary Engine Wrist Pins (high speed) 

f (slow speed) 
Stationary Engine Main Bearings i for dead load.* 

[ for steam load . 

Stationary Engine Crank Pins (slow speed) 

Stationary Engine Wrist Pins (slow speed) 

Gas Engines, Main Bearings 

Gas Engines, Crank Pins 

Gas Engines, Wrist Pins 

Heavy Line Shaft Brass or Babbitt Lining 

Light Line Shaft Cast Iron Bearing Surfaces 

Generator and Dynamo Bearings 



Allowable Bearing Pressure in 


lbs. per 


Square Inch. 


7000 


to 


9000 


3000 


to 


4000 


3000 


to 


4000 


1500 


to 


1700 


190 


to 


220 


300 


to 


3 2 5 


275 


to 


400 


400 


to 


500 


400 


to 


500 


60 to 


120 


150 


to 


250 


900 


to 


1500 


400 


to 


600 


1000 


to 


1800 


80 to 


140 


200 


to 


400 


800 to 


1300 


IOOO 


to 


1500 


500 


to 


700 


1500 


to 


1800 


1500 


to 


2000 


100 


to 


150 


15 


to 


25 


3° 


to 


80 



considerations, it is evident that the length of the journal must 
be such that the bearing pressure is within the allowable limit. 
It may be, however, that the length of the journal thus deter- 
mined will be so great that localized pressure may result; or it 
may be that the type of machine will not allow space enough for 



* Weight of shaft, flywheels, etc. 



252 MACHINE DESIGN 

such a length of bearing. In such cases the diameter must be 

made larger and the length may be correspondingly decreased. 

While practice shows wide variations, it is found that the ratio 

of the length of the journal to its diameter (— J is fairly well 

defined for any given class of machinery. It often occurs, there- 
fore, that, when journals are designed with the ratio as fixed by 
practice, they have an excess of strength while barely satisfying 
the conditions as to bearing pressure. 

The following are average values of — as found in good 

practice : 

TABLE XIV 



TYPE OF BEARING 


Values of -r 
a 


Marine Engine Main Bearings 

Marine Engine Crank Pins 


i to i . 5 
1 to 1 . 5 
i| to 2.5 

1 

1 to 1 . 5 

2 to 3 

3 to 4 

3 


Stationary Engine Main Journals 

Stationary Engine Crank Pins 

Stationary Engine Crosshead Pins 


Ordinary Heavy Shafting with Fixed Bearings .... 
Ordinary Shafting with Self-adjusting Bearings . . . 
Generator Bearings 





99. Summary. From the foregoing the following statements 
may be made regarding imperfectly lubricated journals : 

(a) The minimum diameter of a journal is fixed by the con- 
siderations of strength and stiffness under the loads applied. 

(b) The smaller the diameter of the journal for a given co- 
efficient of friction, the less is the work of friction and consequent 
liability to heating. 

(c) The tendency of the bearing to heat, other things equal, 
is not materially affected by changing the. diameter of the jour- 
nal, but is reduced by increasing the length. 

(d) The projected area of the journal must be such that the 
bearing pressure will be kept within the allowable limits for the 
particular conditions; and the ratio of length to diameter must 
not be so great that severe localization of bearing pressure is 
liable to result. These considerations may require a larger 
bearing than the previous requirements alone would demand. 



CONSTRAINING SURFACES 253 

(e) The work of friction, per unit area, must not exceed the 
rate of radiation, per unit area, for the allowable bearing tempera- 
ture. 

100. Perfectly Lubricated Journals. It was shown in Article 
33 that if a journal is supplied with sufficient lubricant, of 
proper viscosity, the journal itself may draw in the lubricant till 
a film is formed under such a pressure that the load will be en- 
tirely fluid-borne. With any given set of conditions, therefore, 
and perfect lubrication, a definite journal velocity will permit 
the carrying of a definite load per unit area upon the journal, 
and once the relation is established between the load, velocity, 
and coefficient of friction, it is constant, and not unstable, as 
in the case of imperfectly lubricated surfaces. 

It was further shown that the following statements are true 
regarding perfectly lubricated surfaces. 

(a) The friction of perfectly lubricated surfaces for a given 
velocity depends very little on the materials which form the 
rubbing surfaces, but does depend largely on the character of 
the lubricant. 

(b) The frictional resistance of perfectly lubricated journals 
for any given velocity is, within the limits of pressure under which 
the oil film may be maintained, independent of the pressure; 
(that is, p- w = a constant) . 

(c) The coefficient of friction of perfectly lubricated surfaces, 
for any given pressure, varies very nearly as the square root of 
velocity, for velocities up to 500 ft. per minute; approximately 
as the fifth root of the velocity for velocities between 500 and 
2,000 ft. per minute; and is practically independent of the velocity 
for values above 2,000 ft. per minute. 

The first of these statements has been abundantly verified 
by experiment and is discussed fully in Article ^. 

The following table, which is one of the several given in 
Tower's report, shows clearly the truth of the second statement, 
for the frictional resistance is seen to remain practically constant 
with all loads at any fixed velocity. The frictional resistance 
is also seen to vary very nearly as the square root of the velocity. 
Table VIII, Article 1,1,, which was deduced from Table XV, shows 



254 



MACHINE DESIGN 



the coefficients of friction for the range of pressures and velocities 
given, the latter not exceeding 500 ft. per minute. 



TABLE xv 

BATH LUBRICATION 



^.StJ Frictional resistance in pounds per square inch of projected 


area of bearing sur- 




face = 


M. w, for velocities in feet per minute as below. Temperature 


= 90° F. 


• S BI b S 










■o^ rt 


















0.0 « i< 


D 5 ft. 


157 ft. 


209 ft. 


262 ft. 


314 ft. 


366 ft. 


419 ft. 


47 1 ft. 


573 




•583 


.62 


.678 


.721 


•758 


•794 


Seized 


520 




.496 


•546 


•597 


.648 


.691 


•735 


.771 


4i5 




.386 


•445 


•495 


•539 


.582 


.619 


•655 


363 




.306 


•35 


.401 


•444 


.488 


•53 2 


.561 


258 


277 


•357 


.416 


•459 


•5°3 


•547 


•583 


.626 


iS3 


248 


.306 


•364 


.408 


•459 


.510 


.561 


.605 


100 


277 


•357 


•423 


•5°3 


•576 


.619 


.663 


.714 



Tower's experiments at different temperatures show that the 
coefficient of friction, for the above range of pressure and veloci- 
ties, decreases as the temperature increases. His principal experi- 
ments, from which Table XV is taken, were conducted at 90 F. 
and without artificial means of cooling the bearing. The differ- 
ence between the coefficients of friction obtained at 9o°F. and 
those obtained at temperatures as high as are usually allowed in 
practice, can be neglected, as far as designing is concerned, 
especially since those at 90 are on the safe side. Tables VIII and 
XV may, therefore, be taken as representing fairly well the rela- 
tion existing between pressure, velocity, and frictional resistance for 
this range, which fortunately covers the most usual conditions 
in practice. It is to be noted that at the greatest pressure and 
highest velocity, the bearing seized, indicating that with such 
velocity a lower pressure must be assigned, if a perfect oil film 
is to be maintained, or that with this greatest load a lower velocity 
must be assigned, if the bearing is to radiate the heat of friction. 
The work of friction at 471 ft. per minute with a load of 520 lbs. 
is seen to be .771X471=365 ft. lbs. per minute, or about 6 ft. 
lbs. per second. From curve I, Fig. 84 (a), it appears that to 
radiate this amount of energy, the bearing must attain a tempera- 



CONSTRAINING SURFACES 255 

ture of over no° F. above that of the surrounding atmosphere, or 
a total temperature of at least i8o°F. 

It is to be especially noted that within the limits of pressure 
where a perfect oil film will form, the frictional resistance, for a 
given velocity, is practically constant and independent of the 
pressure. (See Table XV.) 

The frictional resistance, and coefficient of friction, for bear- 
ings running at velocities of over 2,000 ft. per minute with perfect 
lubrication, have been quite fully determined by Wm. O.Lasche.* 
The experimental work was very extensive, the results very con- 
clusive, and should be carefully read by designers of high-speed 
machinery. A discussion of these experiments is beyond the 
scope of this treatise, but a few of the most important results will 
be considered. Lasche found that at these high velocities the 
coefficient of friction was practically independent of the velocity, 
but varied inversely with the pressure as in the Tower experi- 
ments, and also varied inversely with the temperature. He found 
that if w be the bearing pressure in pounds per square inch, and / 
the temperature of the bearing in Fahrenheit degrees, then 

tiwif - 32) - 51.2 ..... (2) 

5 1 - 2 , v 

or & w = (7) 

('-32) K6J 

For velocities between 500 and 2,000 ft. per minute the co- 
efficient of friction varies about as the fifth root of the velocity, as 
shown by the experiments of Stribeck. As far as designing is con- 
cerned, the difference between the coefficients for this range and 
those found by Lasche for the higher velocities, may be neglected, 
and Lasche's equation may be applied, without serious error, to 
all velocities above 500 ft. per minute. 
Equation (3) may be written 

xr 5 1 - 2 V , x 

*^ = (7Z^). (4) 

where V is the rubbing velocity in ft. per minute. Since p w V 
is the frictional loss per unit of projected area in foot pounds per 
minute, equation (4) may be used to compute the heat which a, 

* See Traction and Transmission, January, 1903. 



256 MACHINE DESIGN 

perfectly-lubricated high-speed journal can radiate per square 
inch of projected area, and not rise above a temperature t. 

The limiting values of the pressure under which a perfect oil 
film can be maintained, at these high velocities, have not been 
fully determined. In Lasche's experiments a load of 213 pounds 
per square inch of projected area was carried at a velocity of 
1,968 ft. per minute. In Kingsbury's* experiments loads from 
80 to 86 pounds per square inch were repeatedly carried at velo- 
cities up to 1,990 ft. per minute. In both Kingsbury's and 
Lasche's work either the oil circulation, or the bearing it- 
self, was artificially cooled, thus materially assisting the radia- 
tion. 

The values given by these experiments were obtained on 
experimental machines and may be looked upon as limiting 
values. Successful practice in the design of steam turbine bear- 
ings gives velocities ranging from 1,800 to 3,000 ft. per minute, 
with pressures inversely as the velocity ranging from 80 to 50 
pounds per square inch. Where the pressure is as high as 90 
pounds per square inch, it is found that the velocity must be 
kept below 1,800 ft.f per minute. The empirical equation 
w V = 150,000 is much used for this class of work, and gives 
values agreeing with those just quoted. It is evident that with 
these high velocites the radiation must be assisted. Thus let V 
= 2,000 and w = 75 in accordance with the empirical rule just 
given, and let ;t be required to keep the temperature t at 150 F. 
or a temperature of say 75 F. above the atmosphere. 

Then by equation (4) the frictional work is, 

51 . 2 X 2,000 
fi w V = —, r- = 867 ft. lbs. per minute or 14. s ft. lbs. 

(i5o — 3 2 ) 
per second, whereas the bearing alone, if connected to a heavy 
iron frame, will, from Curve 2, Fig. 84(a), radiate only 6.4 ft. lbs. 
per second. Since the specific heat of both water and oil are 
known the supply of either necessary to carry off the excess 
heat of friction can be calculated. 

* Transactions A. S. M. E., Vol. XXVII, page 425. 
t See " Steam Turbines," by Frank Foster, page 181. 



CONSTRAINING SURFACES 



257 



It is to be noted that with perfect lubrication the product a* w 
for any velocity is a constant quantity. It follows therefore that 
for any given total load W the unit bearing pressure should be 
kept as high as possible provided it does not exceed the max- 
imum allowable value for the given conditions. For if the unit 
bearing pressure is decreased, either by increasing the diam- 
eter or length of the bearing, the coefficient of friction is cor- 
respondingly increased; hence the total frictional resistance p W 
is also increased. Care should, of course, be exercised in any 
case that the heat of friction is properly carried away. • 

10 1. Examples of Journal Design. Journals generally form 
an integral part of a shaft or spindle, and the determination of 
the stresses acting upon them is a part of the solution of the 





Fig. 85. 



Fig. 86. 



stresses in the shaft itself. It is desirable, however, to point out 
some of the special features of journal design. 

The actual distribution of pressure over a journal, in the 
direction of the axis, is not known; but there is every reason to 
believe that the distribution is fairly uniform. Thus bearings, 
as a rule, wear quite uniformly over their entire length, where 
fair alignment is maintained. It is customary, in the absence 
of exact data, to assume for computations as to strength and 
rigidity that the load on the journal is concentrated at the 
middle of its length. This assumption is on the safe side, and 
will sometimes give shaft diameters excessively large as far as 
strength is concerned. 

The following examples (a, b and c) show the most important 
cases of journal design. It is assumed in each case that the bear- 

l 7 



258 MACHINE DESIGN 

ings are imperfectly lubricated, which is the most common con- 
dition, but the application of the theory to perfectly lubricated 
journals is obvious. 

Example (a). This case is illustrated in Fig. 85. Here the cen- 
tre of the bearing is fixed at O, by the construction of the machine. 
The centre line of the pulley M is also fixed at XX, by the location 
which the belt must occupy, so that the pulley overhangs the 
bearing by the distance a. The diameter of the pulley d is 40 
inches, a = 10 inches, the pull on the tight side of the belt is 500 
lbs., and the pull on the slack side is 300 lbs. It is required to 
determine the dimensions of the journal. 

The stresses induced in the journal are, torsional stress due 

d 
to the twisting moment (7\ — T 2 ) — , flexural stress due to the bend- 

2 

ing moment (T 1 -f T 2 ) a, and shear due to the direct pull 7\ + T 2 . 
The last is small and is usually neglected (see Art. 26), and the 
journal may be considered as subjected to a combined bend- 
ing and twisting moment. Formula K 2 or K 3 (page 49), there- 
fore, applies. 

The bending moment M = (7\ + T 2 ) a = (500 + 300) 10 = 8,000 

d 
The twisting moment T = (T 1 — T 2 ) — = (500 — 300) 20 = 4,000 

M 8,000 . , . _/>, n . . , 

Hence — = = 2 = x and taking — = . 8 it is found from 

T 4,000 p 

Figure 9 that equation K 2 applies 



. • . M = - [x + vV + 1] T = - [ 2 + V2 2 + 1 ] 4,000 = 8,480 
* 2 2 

pi pnd 3 

From equation J, page 94, M e = — = . 

e 32 

or j.3L^.. 3»x 8,480 

7z p n x 10,000 

. *. d = 2^8 inches (nearly), or say 2%" 

If the length of the bearing be taken at 7 inches (see Table XIV) , 

mi 1 T x + T 2 500 + 300 

the bearing pressure will be — ; =*= = 50 lbs., 

5 - F 2X X 7 15-75 

which is a safe value. 



CONSTRAINING SURFACES 259 

If the number of revolutions be 300 per minute, and fi be 
taken as .015, the work of friction per unit of projected area will 

7T X 2 2^ 

be 50 X .015 X 300 X ' — = 133 ft. lbs. per min., or 2.2 

ft. lbs. per sec. From Curve 1, Fig. 84 (a), it is seen that to radiate 
this amount of energy the temperature of the bearing will rise 
about 50 above the surrounding air. This is a safe value and 
the design is satisfactory. 

Example (b). Let the line of action of the load pass through 
the centre line of the journal, as in the case of the steam-engine 
crank pin in Fig. 86. Let the length of the crank be 18 inches, 
and the total maximum pressure on the crank pin be 25,00c 
pounds. What should be the dimensions of the crank pin in 
order to be safe against rupture and overheating? 

Referring to Table XIV, it is seen that journals of this charac- 
ter are short compared to their diameter, and hence are usually 
strong enough and stiff enough if designed for a bearing pressure 

/ 
low enough to prevent overheating. Let — be taken as 1.25. 

From Table XIII it is seen that 900 lbs. per square inch may be 
safely carried on this type of pin. If d be the diameter of the 
pin and / the length, then the projected area of the pin is lxd = 
1.25 dxd=i.2$d?. 

Whence 1.25^ X 900 = 25,000 
or d? = 22.2 
. * . ^ = 4.7 or say 5 inches, 
and / = 5X1.25 =6.25 inches. 

The pin may now be checked for strength. In a short pin of 
this kind it is more accurate to assume the load uniformly dis- 
tributed along the pin, than to assume it as concentrated at the 
middle. The pin may, therefore, be considered as a cantilever 
uniformly loaded with a load W = 25,000. 

Whence from Table I, case 3, the maximum bending moment 

Wl 25,000 X 6.25 n . , 
M = = = 78,125 inch pounds 



260 MACHINE DESIGN 

pi Me xiM 

. ' , from equation J, page 94, M = — or p = ~j— = — -=■ 

or p = ^ — = 6,400 pounds nearly, which is a safe 

value. 

In a similar way the pin may be checked for deflection, if 
desired, by means of case 3, Table I. 

Example (c) . Sometimes the location of the bearing is depend- 
ent on the diameter of the shaft, which is unknown, and in such 
case a tentative method must be adopted. Thus in Fig. 86 neither 
the length of the bearing B, nor the thickness of the crank hub /, 
can be definitely decided upon till something is known about the 
diameter of the journal. The diameter must therefore be as- 
sumed, and then checked by the equations which apply to the 
case. Usually a close estimate can be made from existing ma- 
chines of similar type. In the case of the steam-engine shaft, 
for example, it is known that the main journal is frequently about 
one-half the diameter of the cylinder. The data taken in example 
(b) correspond to a cylinder diameter of about 18 inches, and the 
journal diameter may therefore be assumed as 9 inches. From 
Table XIV, the length of the journal may be taken as 20 inches. 
The length of the hub should be at least 8 inches, for this di- 
ameter. The boss under the pin may be taken as y% in height 
and since the pin, from case (b), is 6.25 inches long, the total 
distance from the middle of the crank pin to the middle of the bear- 
ing may be assumed as 21X inches. The projected area of the 
journal is 9X20 = 180 square inches, which gives a bearing 

25,000 
pressure of — - — = 140 pounds per square inch; and from Table 

XIII it is seen that this is a safe value as far as the load due to steam 
pressure is concerned. If the shaft also carries a heavy fly-wheel 
this must be taken into account (see next chapter). 

The stresses induced in the journal are of the same character 
as in case (a). Taking the length of crank / = 18 inches, and the 
pressure on the pin = 25,000 as before, then the bending moment 
M = 25,000 X 21 K = 53 7? 500 inch pounds, the twisting moment 



CONSTRAINING SURFACES 261 

M 
T = 25,000 X 18 = 450,000 inch pounds, whence — = 1.19 = 

/>, 
x, and taking — : = .8, equation K* is found by Fig. 9 to apply to 

P 
the case. 

1 



. * . M e = - [x + vV + 1] T = - - [1 . 19 + \/i . 19 2 + 1] 450,000 
•^ 2 

= 616,500. 

17 7 / • 1 \ * $ 2M * 3 2 X 616,500 

From/ (as in example a) p = =- = ^ — = 8,600, 

n d 3 7T X 9 

which is a safe value and the design is satisfactory. 

102. Lubrication of Journals. The point of application of 

the lubricant is of utmost importance, and the method of supplying 

the lubricant to the journal sometimes materially affects the 

design of the bearing. The most common methods of feeding 

lubricants to rubbing surfaces as given in Article 30 apply fully to 

journals and may be classified as follows: 

Common oil hole. 
Common wick or siphon feed cup. 
Common drop sight feed cup. 
Oily pad against journal. 
Ring or chain feed. 
Centrifugal oiler. 
Compression grease cup. 

Bath lubrication. 
Perfect Lubrication \ Flooded lubrication. 
Forced lubrication. 



Imperfect Lubrication 



In flooded lubrication (sometimes erroneously called forced 
lubrication) , the oil is supplied to the bearing under a low pressure 
which insures that the journal is always flooded at the point of 
application, as in bath lubrication, but it does not force the 
lubricant between the surfaces. In forced lubrication the oil is 
supplied at a pressure in excess of the film pressure at the point 
of application, and is thus forced in between the surfaces, no 
reliance being placed on the tendency of the journal to draw in 
the lubricant. The compression grease cup, while supplying 
the lubricant under slight pressure, gives xmly imperfect lubrica- 



262 MACHINE DESIGN 

tion as the supply of lubricant is not copious as in the case of 
forced lubrication. 

In applying any of these methods of lubrication, therefore, 
except the compression grease cup and forced lubrication, care 
should be exercised that the point of application is at, or near, a 
point of lowest pressure and at the place where the journal will 
naturally draw in the lubricant. Thus, in Fig. 80, if the pres- 
sure is always downward, lubricant can be supplied at H for 
motion in either direction. If the pressure were upward, an 
oil hole at H would not only be useless for supplying lubricant 
but would be fatal to good lubrication, as any tendency for a 
pressure film to form would be destroyed by relief of the pressure 
at the hole. In such a case the lubricant should be supplied 
from underneath, or if the direction of rotation were anti-clock- 
wise an oil hole as shown at i" would be good design. In forced 
lubrication the point of application should be the point of greatest 
bearing pressure, and the hydraulic pressure under which the oil 
is supplied should be greater than the maximum bearing pressure. 

While the decreased friction due to perfect lubrication is 
evident, it does not follow that an effort should be made to design 
every bearing so as to secure this advantage. In some places a 
simple oil hole is sufficient, in others a constant supply from a 
wick feed will suffice, while again, with greater speeds, a ring 
oiling device is necessary. In many modern power instal- 
lations, with either steam turbines or reciprocating engines, 
very complete apparatus for supplying flooded lubrication will 
be found. The bearings are constructed so as to catch all the 
oil, as it leaves the journal, and pipes convey it to a central 
receiver. A pump continually circulates the oil to the various 
bearings, and in the best installations the oil is filtered and cooled 
during the circuit. The same results are obtained by flooded 
lubrication as with bath lubrication. , Forced lubrication is 
resorted to only where the bearing pressures are excessive and 
beyond those which can be supported by the natural action of 
the film formed by rotation of the journal. (See Art. 33.) 

The location and character of the oil grooves deserve special 
attention. If the velocity of the journal is so low as to draw in 



CONSTRAINING SURFACES 263 

little lubricant the oil grooves should be so cut as to allow the 
lubricant to flow in near the points of greatest pressure. Grooves, 
or scores on the journal itself, have been found helpful in drawing 
in the lubricant under such circumstances; especially where the 
lubricant is heavy. But where the velocity is above 25 feet per 
minute (see Fig. 16), and for ordinary pressures, care should be 
used that no oil grooves are cut that will tend to prevent the 
formation of the pressure film. If the lubricant is delivered at 
H, Fig. 80, and the pressure is downward, oil grooves of any 
kind running from H which will distribute the oil over the surface 
of the journal, are allowable so long as they terminate at a little 
distance from the edge of the bearing. If the oil is delivered at 
/, and the pressure is either downward or upward, the grooves 
should be cut at right angles to the direction of motion, so as to 
distribute the oil along the entire length of the bearing. If cut 
diagonally they will extend under the journal toward the point of 
greatest oil pressure, thus relieving any tendency to the formation 
of a pressure film and the lubrication will not be as good as it 
would be if no grooves were present. 

The sharp edges of all oil grooves should be carefully removed 
to facilitate the passage of the oil under the journal. The sharp 
edges of the bearings themselves should also be filed or scraped 
away for the same reason. Where one bearing surface encircles 
nearly one-half of the shaft, as in Fig. 80, the surfaces should be 
relieved for some little distance from the parting line to help 
the wedging action of the oil and to insure the journal against 
side pressure due to springing of the bearing under the load. 
A bearing which binds sidewise will not lubricate properly. 

THRUST BEARINGS 

103. General Considerations. When a shaft is subjected to 
a heavy end thrust, either from the weight of the parts carried 
or on account of the power transmitted, the simple collars which 
are used to prevent end thrust in ordinary shafting will not 
suffice, and bearings of special form, known as thrust bearings, 
must be provided. If the bearing is designed so that the thrust 
is taken on the end of the shaft it is called a step-bearing or 



264 



MACHINE DESIGN 



footstep-bearing. If the thrust bearing must be placed at some 
distance from the end of the shaft it is called a collar bearing. 

104. Step-Bearings. If the motion of rotation is very slow, 
as is the case in swinging cranes and similar work, a simple cast- 
iron step, as shown in Fig. 87, will meet the requirements, even 
if the pressure is heavy. If, however, the velocity is high, this 
simple arrangement will not give good results, even when the 
pressure per unit area is low. It may be assumed, without great 
error, that the unit pressure between the faces of a newly fitted 
step-bearing is uniform at all points. The velocity of rubbing, 
however, is a maximum at the outer edge, and, theoretically, it is 
zero at the geometric centre of the pivot. Since the wear is pro- 
portional to the product of pressure and velocity, it follows that 

the surface will wear unevenly, 
the greater wear taking place 
at the outer edge. This will 
bring a concentrated pressure 
at other points, and heating and 
cutting may result. It is always 
advisable in heavy work, for this 
reason, to remove the wearing 
surface near the center, where 
the motion is slowest, and 
where eventually the greatest concentration of pressure is likely 
to be produced (see Fig. 87). Decreasing the bearing pressure 
by increasing the surface, is effective within limits, since the 
area increases as the square of the diameter while the velocity 
of rubbing increases directly as the diameter. Increasing the 
radius, however, increases the average moment arm of the 
frictional resistance, and hence increases the lost energy. It is 
often better, therefore, to carry a higher bearing pressure, and 
thus keep the diameter of the pivot small. 

If a number of discs are placed between the step, or pivot, 
and the bearing (Fig. 88), they have the effect of reducing the 
relative velocity between adjacent surfaces; and if the rotative 
velocity of the pivot is high, they are very useful as a safeguard 
against cutting; for if abrasion should begin between any pair 




Fig. 87. 



Fig. 88. 



CONSTRAINING SURFACES 



265 



of discs, motion will cease at that point till the lubrication 
became effective again. These washers are usually made alter- 
nately of steel and brass, or some other metal, and the upper and 
lower washers are fastened to the shaft and bearing respectively. 
An oil hole passes through the centre of the washers, and radial 
grooves cut across the faces permit a flow of oil between the 
surfaces, centrifugal action assisting the lubrication. If the top 
of the bearing is connected to the bottom by an oil passage, as 
shown at N (Fig. 88), the centrifugal action will set up a con- 
tinuous circulation of the oil, making the lubrication effective. 
The unit pressure between washers is the same as between the 
shaft and the first washer, but the relative motion between the 
surfaces is decreased and the wear thus reduced. A combination 




Fig. 89. 



Fig. 90. 



Fig. 91. 



of hardened and ground steel washers, alternating with brass or 
bronze washers, makes an effective bearing. Sometimes the 
washers are made lenticular in shape, as shown in Fig. 89, in 
order to allow the shaft automatically to adjust its alignment. 
For very light work the shaft sometimes rests on a pair of hardened 
steel buttons, or a hardened steel ball which runs between hardened 
steel surfaces is introduced. In the submerged step-bearings of 
water turbines the shaft, which is often capped with bronze, rests 
on a lignum vitae step and lubrication is effected by the surround- 
ing water. 

If the outline of a step-bearing be made that of a tractrix* 
(Fig. 90), it is found that the tendency to wear in an axial direction 



* See Church's " Mechanics," page 181. 



266 



MACHINE DESIGN 



is uniform at all points; in fact if two homogeneous flat surfaces 
are rotated together they tend to wear into the form of a tractrix 
as has been proven by experiment. This is, therefore, the correct 
shape, theoretically, for all step-bearings; but on account of th? 
difficulty and expense of machining the surfaces, it is seldom 
used. The tractrix has been called Schiele's Anti-friction Curve 
after the discoverer of the above property. This is a misnomer, 
however, for the friction of a tractrix-shaped step is much higher 
than that of a plain pivot. 

It is evident that the rubbing surfaces of all the step-bearings 

which have been discussed can be 
submerged in an oil bath. The lu- 
brication thus obtained is not to be 
confused with that obtained on hori- 
zontal rotating bearings discussed 
-I] formerly. While centrifugal force 
does drive the oil from the centre to 
the outside, there is little action on the 
part of the surfaces themselves tend- 
ing, on account of its viscosity, to 
draw the lubricant between them, as 
in horizontal bearings. Such lubri- 
cation cannot therefore be looked^ on 
as perfect lubrication although giv- 
ing excellent results. The experi- 
ments of Beaucamp Tower* on a 
steel foot step, three inches in diam- 
eter, gives considerable information on this subject. It was 
found that a single diametral oil groove was better than more, 
and pressures up to 160 pounds per square inch were success- 
fully carried at 128 revolutions per minute. The foot step 
was freely lubricated, and rested directly on the bearing, no 
washers being interposed. At 240 pounds per square inch the 
bearing seized. 

If under heavy loads the maintenance of lubrication is im- 




THRUST BEARING OF CURTISS 
VERTICAL STEAM TURBINE 

Fig. 92. 



♦Transactions of the Institute of Mechanical Engineers, 1891, page 111. 



CONSTRAINING SURFACES 



267 



portant, the lubricant should be supplied at the centre of the step- 
bearing under a pressure such that the metallic surfaces are 
forced apart and the load is fluid-borne. Fig. 92 shows a recent 
form of the step-bearing used on the Curtiss steam turbine. The 
vertical shaft A, which supports the heavy rotating parts of both 
turbine and generator, is carried on the disc B which rotates with 
it. The lower disc C can be adjusted vertically, by means of 
the screw E, and is prevented from rocking on E by the screws F. 
Oil is forced between the discs through the central pipe 2^, forcing 
the discs apart and escaping into the cavity G. The load is thus 
completely fluid-borne and perfect lubrication is maintained. 



>Oil Holes 




Fig. 93. 

The oil passes from G upward through the guide bearing escaping 
atfl". 

105. Collar Thrust Bearings. When the thrust bearing must 
be placed at some distance from the end of the shaft, the shaft 
is provided with collars integral with itself, which bear against 
the resisting surfaces as shown in Fig. 93, which illustrates a 
thrust bearing as used for marine work. In cheap work, or 
where the load is small, a single collar is sometimes used. Occa- 
sionally a series of washers, as in Fig. 88, are interposed between 
the collar and the bearing ring. The objection to the single- 
collar bearing for heavy loads is that the large diameter necessary 
to obtain a practical bearing pressure increases the work of 
friction, due to the increased velocity, and the difference between 
the rubbing velocities of the ring at the shaft and at its outer 
diameter results in unequal wear. The outer diameter of the 
ring, or collar, is usually, therefore, not more than one and one- 



268 MACHINE DESIGN 

half times the diameter of the shaft, which limits the width of 
face of the collar even in large shafts to a few inches; and the 
necessary area is obtained by using a number of rings. 

In small or cheap work, the bearing surfaces of the thrust block 
are sometimes made integral with the bearing proper; but usually 
they are made detachable. Thus the main casting of the block 
may be of cast iron and the bearing rings of brass are inserted and 
held in place by radial grooves cut in the block. These rings must 
be scraped until each collar on the shaft bears properly against 
its mating ring, so that the thrust is uniformly distributed. The 
most modern practice in marine work is to make the bearing rings 
horseshoe-shaped, as in Fig. 91, so that each ring can be with- 
drawn without disturbing any other portion of the bearing or 
shaft. Occasionally the horseshoe collars are adjustable along 
the shaft so as to be more easily brought to a proper bearing. In 
first-class work each horseshoe has its own independent water 
circulation, so that local heating may be prevented, and the lower 
part of the bearing constitutes an oil bath into which the collars 
dip. This oil bath also has a water circulation for cooling the oil. 
106. Friction and Efficiency of Thrust Bearings. If P be the 
total load on a flat circular pivot of radius r 1 and p. be the co- 
efficient of friction, then the frictional moment resisting rotation is 

M = -vPr* . . . •■. . . (1) 
3 

If r t be in inches and P be in pounds then the energy lost per 
minute in foot pounds is 

2 2 n JV 

E = - / ,Pr l X — — = .^guP^N . . (2) 

3 I2 

where N is the number of revolutions per minute. 

In a similar manner if the thrust be taken on a collar of out- 
side radius r lf and inside radius r 2 , then 

*- ri "$Egj • • • (3) 



3 



3 r*- 



mdE=. 349 vPN(^—^ ... (4) 

■* Church's " Mechanics," page 180. 



CONSTRAINING SURFACES 



269 



The efficiency of a thrust bearing cannot always be expressed 
as a function of the power transmitted. Thus in the case of a 
vertical shaft carrying a heavy load of gears, the frictional 
resistance of the step has little to do with the power transmitted. 
In the case of the thrust bearing of a steamship the frictional 
moment and energy loss are directly proportional to the driving 
force P. In either case, however, the frictional moment or the 
energy I09S must be added to the turning moment or the energy 
supplied, as the case may be. 

The following coefficients of friction are taken from Tower's 
experiments : 

TABLE XVI 



Pressures in lbs. per 
Unit Area. 


Coefficients 


of Friction of Flat Pivots for the Revolutions per 
Minute as given below. 


So R.P.M. 


128 R.P.M. 


194 R.P.M. 


290 R.P.M. 


353 R.P.M. 


20 

40 

80 

120 

140 


.0196 
.0147 
.0181 
.0221 


.0080 
.0054 
.0063 
.0083 
.0093 


.0102 
;oo6i 
.0045 
.0052 
.0062 


.0178 
.0107 
.0064 
.0048 
.0046 


.0167 
.0096 
.0063 

•°053 
.0054 



At 50 and 128 R.P.M., the oil supply was restricted, but at 
the other velocites the bearing was flooded. In all cases the 
coefficient increased at revolutions below 40 R.P.M., which 
was probably due to the decrease of the centrifugal force 
(the bearing being oiled from the centre). This would seem 
to indicate that devices for reducing relative rubbing veloc- 
ity, such as multiple washers (Art. 102), may be carried to an 
extreme, causing more friction than a plain flat pivot where 
centrifugal action is effective. In the case of thrust collars, such 
as shown in Fig. 91, running in an oil bath, the surfaces them- 
selves tend to draw in lubricant in a way similar to that of the 
ordinary journal. The coefficients of friction for this class of 
thrust should therefore be as low at least as those given above. 

107. Bearing Pressures on Thrust Bearings. Where the 
velocity of rubbing is very low and wear is not important, as in 
the case of swinging cranes, very heavy unit loads may be put 



270 



MACHINE DESIGN 



upon pivot bearings, especially if they rotate in an oil bath. 
Where the velocity is high, or even moderate, and wear is im- 
portant, much lower pressures must be carried with imperfect 
lubrication, than on ordinary bearings running at the same 
velocity. With forced lubrication, as in the step-bearing shown 
in Fig. 92, it is evident that very heavy pressure may be main- 
tained. If, on the other hand, too many collars are used on a 
collar thrust bearing, in an effort to keep the bearing pressure 
down to a low value, there is danger that all of the collars will 
not bear simultaneously. The following are average values of 
bearing pressures, for thrust bearings, as found in practice : 

TABLE XVII 



Mean Velocity in ft. per Min. 



Character of Lubrication. 



Bearing Pressure in lbs. per 
Square Inch. 



Very slow as in hand cranes 
Up to 50 ft. 
50 to 125 
125 to 200 
200 to 500 
500 to 800 



Bath as in Fig. 87 
Bath as in Fig. 88 
Bath as in Fig. 88 
Bath as in Fig. 88 
Bath as in Fig. 88 
Thrust Bearing and Bath 
Lubrication as in Fig. 93. 



2,000 to 3,000 
200 

150 
100 . 

5o 
75 to 50 



Example. Design the thrust journal for a steamship having 
the following data, and estimate the frictional loss in the thrust 
bearing. 

Speed in knots (1 knot= 6,080 ft. per hour) 15 

Indicated horse-power of one engine 5,000 

Inside diameter of thrust collars 14" 

Outside diameter of thrust collars 2 1" 

Allowable pressure per sq. in. of surface 40 lbs. 

Revolutions of the shaft per minute 1 20 

Owing to frictional losses in the engine, propeller, and shaft 

only about two-thirds of the indicated power is delivered to the 

thrust block. The pressure against the thrust block multiplied 

by the distance through which the ship moves per minute must 

equal the energy delivered to the block per minute; or if P be the 

thrust, S the speed of the ship in knots per hour, and the indicated 

2 
horse-power be denoted by J. H.P., then -Xl.H.P. X 33,000 

o 



CONSTRAINING SURFACES 271 

P X S X 6,080 n 2X I.H.P. X 33,000 X 60 I.H.P. X 217 

_ QJ- p _ ^ _ __ f 

60 3 X ^ X 6,080 5 

1 1 n 5>°°° X 217 
Hence 111 the above example P = = 72,300. The 

n 
area of each thrust collar = — (21— 14 2 ) = 192 sq. in. There 

4 
fore the total allowable pressure on each collar = 192 X 40 = 

72,300 

7,680 and the number of collars = 777- = g.s or say 10. 

' 7 ,680 y J J 

If the bearing runs in an oil bath, the coefficient of friction 
will not be more than .01 under the worst ordinary conditions. 

r> 3 _ r 3-1 
Therefore from (4) E = .349 vP N \_ \ \ 

1 2 

rio.5 3 -7 3 -| 

= •349 X .OI X 72,300 X I20 [_ IQ .2.2 J 

= 405,000 ft. lbs. per min. or 12.3 H.P. 

ROLLER AND BALL BEARINGS 

108. General Consideration of Rolling. It was pointed out 
in Article 29 that the resistance due to rolling friction was much 
less than that due to sliding friction, for a given load. The 
application of this principle to very heavy loads and low speeds, 
as in the case of moving heavy bodies en rollers, is of great 
antiquity; but only in recent years have mechanics been able to 
produce surfaces of such a character as could carry even very 
light loads at high speeds on either roller or ball bearings. At 
present, however, bearings of this character can be obtained which 
will run well under very severe conditions. 

When a curved surface rolls upon any other surface with 
which it theoretically makes line or point contact, the two sur- 
faces tend mutually to deform each other, the amount of deforma- 
tion depending on the character and hardness of the materials 
forming the surfaces, and the intensity of the load sustained. If 
the surfaces of both members are very hard, and the load is very 
light, the deformation is negligible and true rolling can be«practi- 



272 MACHINE DESIGN 

cally attained. When, however, any appreciable load is to be 
carried the mutual deformation of the surfaces destroys the 
theoretical line or point contact and the load is borne on a small 
surface. This occurs even when the surfaces are very hard, and 
the action instead of being that of pure rolling, is a combination 
of rolling and sliding.* The true theory of this action, which is 
very complex, has not been fully demonstrated and is beyond the 
scope of this treatise. It can readily be seen that it is closely 
connected with the elastic properties of materials, on which much 
research work has been done. Undoubtedly the work of this 
character, which is of most value in the design of roller or ball 
bearings, is that of Professor Stribeck whose masterly report has 
been translated into English by Mr. Henry Hess,f and to this 
translation reference will be made hereafter. 

If the intensity of pressure be such that the elastic limit of 
the materials is exceeded, permanent deformation will occur. 
In the case of roller or ball bearings this may result in the destruc- 
tion of the surfaces either by flaking off locally, or by simply 
crushing out of shape. In either case continued action of this 
character is destructive to the bearing. Experiments on either 
balls or rollers to determine the ultimate crushing load are, there- 
fore, misleading and useless as far as the design of such bearings 
is concerned. It appears from experiment and experience that 
bearings of this character can be constructed to carry fairly 
heavy loads at high speeds for a long period of time provided 
the intensity of pressure is not too great. It is obvious from 
the foregoing that the materials used in such bearings must be 
homogeneous, and of uniform hardness. The success of the 
modern ball and roller bearing has been made possible by im- 
proved materials and workmanship rather than by new theories. 

Referring to Fig. 94, it is evident that when two adjacent 
rollers or balls, A and B, touch each other, the directions of 
motion of the common points of contact are in opposite directions. 
It is often stated that this results in considerable frictional loss; 

* The student may demonstrate this action by rolling a round lead pencil on 
a piece of soft rubber under pressure. 

f See Transactions A. S. M. E., Vol. 29, pages 367 and 420. 



CONSTRAINING SURFACES 273 

and sometimes small intermediate balls, or rollers, are used as 
shown at C, Fig. 94, to obviate the supposed loss. Such inter- 
mediate balls or rollers must be kept in place by a cage such as 
E, Fig. 94, and this cage will give rise to a greater frictional loss 
than that which it is expected to remedy. A brief reflection will 
show that very little pressure can possibly exist between A and B. 
The only pressures that can be exerted by the guiding surfaces 
upon the balls or rollers are in a radial direction or normal to the 
surfaces, and these have no component tending to force the ad- 
jacent rollers or balls together. Sometimes the rollers or balls are 
separated by a guiding cage (see Fig. 95) , and if any appreciable 
pressure could exist between adjacent rollers or balls the same 
would necessarily exist between them and this guiding cage. 
This theory is not borne out by experience, as these cages, in 
well-built roller bearings, do not wear appreciably. The fric- 
tional loss from this source is undoubtedly very small. , 

The friction of roller and ball bearings while at rest is very 
small, and this is a very important point in the design of heavy, 
slow-moving machinery where, with ordinary sliding bearings, 
it often takes a much greater effort to start the machinery from 
rest than to maintain motion at full speed. 

ROLLER BEARINGS 

109. Forms of Bearings. Roller bearings, in common with 
the ordinary bearing, are classified as radial or thrust bearings, 
according to the manner in which the load is sustained. A 
typical form of construction of roller bearings for radial loading 
is shown in Fig. 95. A shell of hardened steel, B, surrounds 
the shaft ^4, and is secured firmly to it. The rollers C bear against 
this shell B, and against an outer shell D, which is secured to the 
bearing proper, E. Both rollers and shells are usually made of 
high carbon steel hardened and ground, or of mild steel case- 
hardened. The rollers are held parallel with the axis of the shaft 
by means of a cage F which is made of brass or other soft material. 
Some form of cage is necessary in all roller bearings on account 
of the tendency of the rollers to twist out of line with the shaft, 
thus replacing the theoretical line contact with point contact, and 
18 



274 



MACHINE DESIGN 



also causing an end pressure and cramping on the rollers. This 
tendency to end thrust is sometimes provided for by putting a 
small ball at each end of the roller to act as a thrust bearing. If 
the axis of the roller is not parallel to that of the shaft, it cannot 
make line contact with the shaft unless it assumes a spiral form. 




Fig. 94„ 



If the surfaces which confine the roller are accurately made, and 
the clearance is very small, as it should be, the roller cannot get 
out of parallelism with the shaft without being bent into a spiral 
form. If the rollers are hardened this may result in fracturing 
them, especially if they are relatively long. To obviate this 




FlGo 96. 



Fig. 97. 



trouble the rollers are sometimes made in short lengths, as shown 
at H, in Fig. 95, or the roller is made flexible as illustrated by the 
Hyatt roller shown in Fig. 96. This roller is made by winding 
steel strip spirally upon a mandrel, thus making a hollow flexible 
roller. It is to be especially noted that neither of these methods 
will compensate for inaccurate workmanship. For continuous 



CONSTRAINING SURFACES 



2 75 



line contact the outer and inner shells must be machined with 
great accuracy, placed in very accurate alignment, and the rollers 
must be guided so as to remain perfectly parallel to the shaft. 
These conditions are difficult to obtain initially, and almost 
impossible to maintain with great accuracy under continuous 
service. The rollers in bearings for radial loading may be 
cylindrical or they may be conical as in the Grant bearing shown 
in Fig. 97. The construction here shown permits of adjust- 
ment for wear, which is difficult to obtain where the roller is 
cylindrical. 

If the direction of the load to be carried is axial, roller thrust 
bearings of the form shown in Fig. 98 are often used. The shaft 




Fig. 98. 



Fig. 99. 



A carries a thrust collar B and the thrust is taken on the frame of 
the machine by a corresponding collar C. A hardened steel 
ring D is attached to B and rotates with it, while a similar ring 
E is fastened to the stationary part C. The conical rollers G 
roll between these rings, carrying with them the cage F. A thrust 
ring H prevents the rollers from moving radially outward. The 
apex angle of the roller should not exceed 15 , and in most cases 
is kept down to 6° or 7 to prevent serious end pressure against 
this retaining ring. It is evident that where the roller is conical 
in form, the apex of the cone lying in the centre line of the shaft, 
the velocity of any point in its periphery is proportional to its 
distance from the axis of the shaft and, theoretically, true rolling 
will be obtained. 



276 MACHINE DESIGN 

Bearings of this character with conical rollers are expensive 
to make in an accurate manner, and a simpler form, as shown in 
Fig. 99, is sometimes used. Here the rollers are cylindrical in 
form and are made in short lengths so as to reduce relative 
slipping. The outer rollers rotate faster than the inner rollers, 
and the lengths and arrangement of the rollers are such that 
ridges are not worn in the seat. 

Space does not permit of discussion of the many forms of 
roller bearings on the market; but their fundamental principles 
are the same, and the student is referred to current trade cata- 
logues for variations in methods of construction. 

no. Allowable Bearing Pressures. It is evident that the 
bearing pressure in roller bearings must not be great enough to 
stress the material of either roller or bearing surface beyond the 
elastic limit, but theoretical considerations are of little service in 
the actual designing of such bearings. The most reliable ex- 
perimental data bearing on the subject are the results of 
Stribeck's work. In roller bearings under radial pressure 
the quantity equivalent to the projected area of the ordinary 
bearing, as far as carrying capacity is concerned, is considered 
as the product of length and diameter of a single roller, multi- 
plied by one-fifth the total number of rollers in the bearing. 
Thus, according to Stribeck, for cylindrical bearings, if 
N = total number of rollers. 
W = total load on bearing in lbs. 
w = load on one roller in lbs. 
d = diameter of roller in inches (mean diameter for conical 

rollers) . 
/ = length of roller in inches. 
k = a constant to be determined experimentally. 
Then w = kid . . (1) 

and W = k I d — (2) 

5 
From Stribeck's* experiments k has a value of 550 for unhardened 

rollers and bearing surfaces and 1,000 for hardened surfaces. 

* See Transactions A. S. M. E., Vol. XXVII, page 444. 



CONSTRAINING SURFACES 



277 



In the case of thrust bearings the load may be considered as 
distributed over the total number of rollers. Bearings of the 
type shown in Fig. 99 have been constructed to carry a load of 
156,000 pounds at 250 revolutions per minute. 



BALL BEARINGS 



in. Theoretical Considerations. Let the ball A, Fig. 100 (b), 
roll along the circular path* B, with pure rolling motion, making 
point contact with the path. Let the path B be parallel to the 
plane CD, and suppose also that the ball as it rolls remains a 
fixed distance from this same plane. Then it is evident that if 
A rolls with pure rolling motion along B, it will rotate around 







P 


r 








1 
1 




s/A 


Q 


e- 


\j/\. 




Oi 


6 


















© 










a 










oj 




















fu 






X. \// 


rb > 










D 


C b$^ 






k 



Fig. 100 (a). 



Fig. 100 (b). 



some one of its diameters, at right angles to B, as an axis, and 
will make contact with B along the edges of such a disc as would 
be cut from it by a plane passing through the point of contact 
b perpendicular to the diameter around which the ball rotates. 
Thus the ball may rotate around Ok as an axis, and roll along 
the edges of the disc b i. It is clear, however, that the ball can 
rotate around only one diameter at a time, and preserve true 
rolling contact with B. If the ball has two concentric paths of 
contact as B and E, Fig. 100 (b) whose points of contact with the 
ball are b and e (Fig. 100 a) respectively, then it must roll along 
two discs b-i and e-l, and these discs must have a common axis 
of rotation O k perpendicular to their planes and passing through 
the centre of the ball. Further, the discs must be so placed that 



* The guiding surfaces of ball bearings are almost invariably circular in form. 



278 MACHINE DESIGN 

the lines i I and b e intersect on the line m, passing through the 
common centre of B and E; for then 

p e rb 

or the circumferences of the rolling discs are proportional to the 
circumferences of the paths of contact, and true rolling may be 
attained. It is not possible to have more than two points of 
contact between the ball and one of its guiding surfaces, with 
pure rolling, as the proportionality given above is not true for 
any other points on the line b except those given. The above 
principles are fundamental and apply to all ball bearings with 
circular guiding surfaces. 

112. Spinning. Usually one of the guiding members is fixed 
and the other rotates, the friction between the moving member 
and the ball causing the latter to roll. If the load carried is so 
small that no distortion of the surfaces takes place, and true point 
contact exists, this frictional force will act tangent to the outer 
circumference of the disc of contact and be parallel to its plane. 
Such theoretical conditions never exist in practice, as the sur- 
faces of contact are deformed, even under light loads, and the 
load is carried on a small area instead of a point. The fric- 
tional force rotating the ball is, hence, indeterminate and in 
general has components which tend to rotate the ball about 
other axes than the one which will give pure rolling motion. 
It is clear that inaccurate workmanship will give the same result. 
This action is known as spinning and is necessarily accompanied 
by friction. 

113. Forms of Bearings. Ball bearings are divided into three 
types, according to the character of the load and the way it is 
sustained by the bearing: 

(a) Radial bearings, for loads acting at right angles to the shaft. 

(b) Thrust bearings, for loads acting parallel to the axis of 

the shaft. 

(c) Angular bearings, for taking loads both perpendicular 

and parallel to the axis of the shaft. 



CONSTRAINING SURFACES 



279 



Each of these types maybe either a two-point, three-point or four- 
point bearing, depending on the number of points of contact made 
by the ball on the guiding surfaces. 

114. Radial Bearings. Figure 1 01 (a) shows a two-point radial 
bearing. The race B is secured to the shaft A, while the race F 
is secured to the other member C. Either A or C may be the 
rotating part. In order to place the balls in the raceway an 
opening is often cut in the side of one of the races, as shown at E, 
and the opening then closed with a filling piece as shown. If the 
race F is stationary this filling piece can be located on the un- 
loaded side and no wear brought upon it. If B is stationary the 
opening must be cut in it, and the same care used in locating the 




Fig. 10 i (a). 



Fig. ioi (b). 



filling piece with reference to the load. If both the shaft A and 
hub C rotate this cannot be accomplished, and the full load is 
brought upon this filling piece, thus decreasing the capacity of the 
bearing to sustain a load, on account of the break in the surface 
of the race. If the velocity of the rotating member is high this 
break in continuity of the race is destructive to the bearing. 

If about half the total number of balls necessary completely to 
fill the race is used, each race may be made of one solid piece. 
In such cases the bearing is assembled by moving the inner race 
over eccentrically to the outer race, filling in the balls and then 
distributing them. Separators of elastic material are then pushed 
in between the balls to maintain correct spacing. These separat- 
ors, also, often act as reservoirs for lubricant. They may be of 



280 



MACHINE DESIGN 



felt or such soft material or may be made in the form of a helical 
spring. This construction is shown in Fig. 1 01 (b). The lessened 
number of balls is compensated for by using balls of larger 
diameter and hence greater carrying capacity. 

The carrying capacity of radial ball bearings, according to 
Stribeck's experiments, is not affected materially by velocity, 
within reasonable limits, so long as the velocity of rotation is 
uniform ; but sharp variations of velocity at high speed reduce 
the capacity. 

115. Thrust Bearings. Fig. 102 illustrates a four-point thrust 
bearing. Here there is no difficulty in filling in the balls when the 
races are solid. In Fig. 102 the angles <f> and $' are equal, but 
this is not necessary as it is evident that any line drawn through 



D v B 






Fig. 102. 



Fig. 103. 



O and intersecting the ball circle will locate a pair of rolling discs 
which will roll on B, without interfering with the pair shown, 
which may roll on A. 

The surfaces C and D are sometimes made both flat and 
parallel. It is difficult, however, to obtain absolute parallelism, 
initially, between C, D and the ball races, and much more difficult 
to maintain this parallelism under running conditions. An error 
in alignment, either from poor workmanship or deflection under 
load, of less than one thousandth of an inch will cause concen- 
trated loading of the balls on one side. If possible, therefore, 
such bearings should be seated on spherical surfaces, as shown 
at Z>, thus allowing the races to adjust themselves correctly. 



CONSTRAINING SURFACES 



28l 



Mr. Henry Hess states that speed is an important factor in such 
bearings and he gives 1,500 revolutions per minute as a max- 
imum. 

A simple form of ball thrust bearing is shown in Fig. 103. 
Here the balls run against flat hardened surfaces, A and B, and 
are kept in position by a cage C made of some soft alloy. The 
cage may be made to retain the ball loosely by drilling the open- 
ings for the balls almost through as shown in Fig. 103 (b), in- 
serting the ball and then closing down the upper edge a little with 
a set as shown at e, Fig. 103 (b). 

116. Angular Bearings. If possible, radial loads should be 
supported by radial bearings, axial loads by thrust bearings, 
and angular bearings should be avoided. Radial bearings should 




Fig. 104. 



not sustain heavy axial loads and thrust bearings should not be 
loaded radially. For light loads the angular bearing will sustain 
pressure in either of these directions. There are innumerable 
forms of angular bearings. Fig. 104 (a), (b), (c), and (d) may be 
taken as typical of two-, three-, and four-point angular bearings. 
The races can be made continuous in all cases, and are often ad- 
justable. This last feature, while sometimes necessary and often 
claimed to be an advantage, is really a detriment as it puts the 
bearing at the mercy of an unskilled person. Properly designed 
ball bearings do not wear appreciably, and if wear does take 
place it will occur on the loaded side only; and adjustment cannot 
compensate for this, but only hastens the failure of the bearing. 
It is evident that all the arrangements shown in Fig. 104 



282 MACHINE DESIGN 

fulfill the requirements for pure rolling contact as outlined in 
Art. 109. The path of the ball is not so definitely determined 
at a, Fig. 104, as in the other forms. For this reason the radius 
of the ball races should, in order to prevent wedging of the ball, 
not be greater than three-quarters the diameter of the ball. For 
the same reason the angle <j> in Fig. 104 (b) should not be less than 
about 25 . In Fig. 104 (b) and 104 (c) the point a may, theoreti- 
cally, be anywhere, as long as it lies between the discs which roll 
on the outer raceway. It should be so placed, however, as 
nearly to equalize the loads at b and c. 

117. Allowable Load. The allowable load which may be 
put upon a ball bearing will depend on the following: 

(a) The character of the materials forming the balls and races. 

(b) The shape of the raceways. 

(c) The diameter of the balls. 

(a) Ball bearings fail by over stressing the material of the 
raceways or balls. If the stress induced is far beyond the elastic 
limit, and often repeated, the surfaces will flake off and failure 
will occur. Experiments on the crushing strength of balls or 
races are useless and misleading as the life of the bearing depends 
on the elastic and not the crushing strength. Evidently none 
but hard materials can be used for appreciable loads and these 
must be homogeneous in texture. Case-hardened materials 
are of doubtful value for severe service. For most trying cir- 
cumstances special steels and alloys will no doubt be much used. 

(b) Theoretically, a ball supports the load on a point, but 
practically the unavoidable distortion of the material increases 
the point to a small surface. It can be demonstrated mathe- 
matically, and is evident on reflection, that a greater bearing 
surface will be formed for a given distortion of ball and ball race 
the more closely the cross-section of the ball race corresponds to 
the cross-section of the ball. On the other hand, and as a di- 
rect consequence of this increase of surface, it is found that the 
friction increases as the cross-section of the races approaches the 
cross-section of the ball, a result to be expected. 

It is almost impossible to machine and adjust ball bearings 
of three- or four-point contact so that the load is uniformly dis- 



CONSTRAINING SURFACES 283 

tributed at the various points of contact. It is borne out by 
experiment and it is well known that two-point bearings can carry 
heavier loads, than any other form for a given diameter of ball. 

(c) The allowable load which a ball can carry varies with the 
square of the diameter. 

These statements have been proven experimentally by 
Stribeck, who found that the carrying capacity of a ball could be 
expressed by w-=k#. m (1) 

where w = greatest load on one ball in pounds. 

k = a constant depending on the material and shape of 

ball races. 
d = diameter of ball in inches. 

Stribeck shows that the total load which may be carried on a 
single-row ball bearing is equal to one-fifth of the allowable load 
on one ball multiplied by the number of balls. If, therefore, W 
be the total load in pounds on one row of balls, and N the total 
number of balls, 

W = w — = k d? — (2) 

5 5 

For hardened steel races made of good quality of steel 

k = 450 to 750 for flat or conical races, three- or four- 
point contact. 

k =1,500 for two-point contact and raceways whose 
radius of curvature equals % d. 

With more perfect materials Stribeck states that these values 
may be increased fifty per cent. 

118. Practical Considerations. It is clear that in order to 
insure an even distribution of load, initially, the workmanship 
on both balls and races must be very accurate; and in order to 
maintain this distribution the material must be uniform in quality 
and hardness throughout. It is also found that, for best results, 
the surfaces must be highly polished and free from scratches. 
The bearing must be kept free from acid and rust and provision 
made for excluding dust and grit and for retaining a supply of 
lubricant, the function of the lubricant being largely to prevent 
rusting. 



284 MACHINE DESIGN 

As before stated, it has been found better to carry the load on 
one row of balls, if possible. Where this cannot be done special 
provision should be made to insure that each of the several rows 
of balls carry its proportionate load. This usually leads to 
some form of equalizing-device which complicates the design. 

The minimum diameter of the shaft is fixed, approximately, 
by the load carried, and balls are made, commercially, in stand- 
ard sizes. In designing a bearing for a full number of balls a 
tentative computation must generally be made to fix the proper 
number and diameter of the balls. Knowing these, the exact 
diameter of the circle passing through the centre of the balls can 
be determined as follows : 

Referring to Fig. 101 (a), draw a line connecting the centresof 
any two balls in contact as G and H, and draw the radii O G, 
O H and O i, as shown. Also let r be the radius of the ball and R 
the radius of the ball circle. 



Then r = 


R sin 


and = 


180 

N 


.'. R = 


r 


. 180 

sm — r—- 



N 

This value must be increased sufficiently to allow for the necessary 
clearance. 

The methods of applying ball bearings are so numerous and 
varied, that no attempt can be made here to illustrate them, and 
the student is referred to the following sources of information on 
this point : 

Transactions of A. S. M. E., Vol. XXVII and Vol. XXVIII. 

Trade publications generally. 



CHAPTER XI 
AXLES, SHAFTS, AND SHAFT COUPLINGS 

119. General. The terms axle, shaft, and spindle are applied 
somewhat indiscriminately to machine members which are so 
constrained by journals and bearings as to admit of motion of 
rotation. These rotating members may be subjected to simple 
torsion or bending, or to combinations of torsion and bending. 
Shear, also, usually exists as in the case of loaded beams. 
Rotating members may be classified roughly as follows, according 
to the predominating stress (see Art. 26), or to the particular 
purpose for which they are intended. 

(a) Axles, loaded transversely and subjected principally to 
bending. 

(b) Shafts, subjected to torsion or combined torsion and 
bending. 

(c) Spindles, or short shafts which directly carry a tool for 
actually doing work, and which as a consequence must have 
accurate motion. 

The axles of railway freight cars are good examples of case 
(a); transmission shafting in factories, or the shafts of steam 
engines are good examples of (b) ; while lathe and milling-machine 
spindles illustrate (c). 

Considerations of strength seldom enter into the design of 
spindles. In these members torsional stiffness and accuracy of 
form in the bearings are, usually, the most important considera- 
tions. When the spindle is designed with these latter require- 
ments in view, there is usually an excess of strength against 
rupture. The discussions given in Art. 12 apply in this case, 
and it will not be considered further here. 

120. Axles. Let A (Fig. 105) be an axle which carries the 
loads P v P 2 and P 3 , but is not subjected to any torsional stress 

285 



286 



MACHINE DESIGN 



except that due to negligible bearing friction. Suppose the axle 
to be supported by the bearings N and N. The distribution of 
the bearing reactions is indeterminate, as explained in Art. 95, 
and the assumption is usually made that they are concentrated at 
the middle of the bearings, as indicated. This assumption is 
on the safe side, so far as the strength of the shaft is concerned, as 
the slightest deflection of the shaft tends to concentrate the re- 
action at the inner edge of the bearing. The axle can, therefore, 
be treated as a simple beam (Art. 14). If the load P 2 were zero, 
and the loads P t and P 3 were equal and symmetrically placed 
(which is the most usual condition, as in car axles), the case 




space diagkam 

Fig. 105. 

would be identical with Case XIV of Table I. It will be in- 
structive, however, to make a solution of the general case given 
above. 

The principal stress to which the axle is subjected is simple 
bending. Shear also exists in every section; but from the general 
theory of beams (Art. 14) it is known that, usually, this latter may 
be neglected in the body of the shaft. If, however, the shaft is 
short, and consequently need not be large to withstand the applied 
bending moment, the section of the bearing at XX should be 
checked for shearing stress. The dangerous section of the shaft 
will be where the bending moment is a maximum, and hence 
it is necessary to determine this maximum moment, which also 



AXLES, SHAFTS, AND SHAFT COUPLINGS 287 

involves the determination of the unknown reactions. The 
reactions may be determined mathematically by taking moments 
around R 2 . 

Then, RJ = P 1 / 1 + P 2 l 2 + PJ 3 

and P 2 = P t + P 2 + P 3 - P t 

The bending moment at any section is the algebraic sum of all 
the moments at either side of the plane considered. Thus the 
bending moment at P 2 = M 2 = R l (I — l 2 ) — P x (l t — l 2 ) and 

this value may be used in equation / of Table VI ( M 2 = — ) 

to determine the stress for a given cross section, or to determine 
the cross section for an assumed stress. 

A graphical solution is much more convenient, as it shows 
at once where the maximum bending moment is located. In 
Fig. 105, denote the forces P D P 2 , etc., thus, ab, be, cd, etc., and 
draw the corresponding force diagram as shown, making AB = 
P v BC = P 2 , and CD = P 3 , to any convenient scale. It is to 
be noted that these forces are drawn consecutively downward, 
since they act in that direction, and their sum, AD, must equal 
the sum of the reactions, or vertical forces. Take any convenient 
pole, as O, and draw OA, OB, OC and OD. From any point on 
ab, in the space diagram, draw oa and ob, parallel respectively to 
OA and OB in the force diagram. From the intersection of ob 
and be draw oc, parallel to OC, and in similar manner draw od. 
Join the intersection of oa and ea with the intersection of od and 
de, thus locating the closing string oe. Draw OE parallel to oe, 
locating E. Then in the force diagram DE = R 2 , and EA = R t 
to the assumed scale of the force diagram. 

The vertical ordinates of the space polygon are proportional 
to the bending moments at the points considered. The numerical 
value of any bending moment is the continued product of the 
length of the ordinate, the perpendicular distance of O from AD, 
the reciprocal of the scale of the space diagram, and the reciprocal 



288 MACHINE DESIGN 

of the scale of the force diagram. Thus if the ordinate at some 
point be 2" long, the pole distance be 2^", the space scale be 
1/4" to 1 ft., or yi size, and 1" = 5,000 lbs. on the force diagram; 
then the bending moment at the point considered is M = 2 X 
2% X 8 X 5,000 = 200,000 inch lbs.; and from this moment 
the diameter of the shaft may be computed. 

121. Shafts Subjected Principally to Torsion. The funda- 
mental* relations existing in a shaft which is subjected to torsion 
only have been fully discussed in Article 12, and for such cases 
or where other stresses, such as those due to bending, are negli- 
gible, Article 12 is applicable. Shafts subjected to pure torsion 
rarely occur in practice, as bending is almost always present 
due to the weight of the shaft itself, and to the weight of pulleys 
which it supports, as well as to belt pull, etc. There are many 
cases, however, where the torsional stress is predominant, and 
where the secondary bending effect is difficult to compute. Thus 
in long factory shafting, where the power is supplied to the shaft 
at one point, and is given off in small increments at short inter- 
vals all along the shaft, the bending due to the pull of the belts 
is small. This is especially true if care is exercised to place the 
pulleys as close to the bearings as possible. 

If the shaft is of considerable length, the angular distortion 
is of importance, and it may often occur that a shaft having 
sufficient torsional strength will not have proper torsional stiffness. 
If the power is applied at one end of the shaft, and taken off at 
the other end, computations for both strength and stiffness are 
easily made and may be of service. In nearly all cases, how 
ever, power is delivered in varying quantities all along the shaft, 
and such computations are not only difficult to make but would 
indicate that the diameter of the shaft should vary at different 
parts of its length. This would be undesirable, as it is important 
that shafting, hangers, etc., should, as far as possible, be uniform 
and interchangeable for convenience and economy; and the 
practice of reducing the diameter of the shaft as it extends from 
the driving point is confined to larger shafting (say over 3" in 
diameter). The design of shafts subjected principally to torsion, 
therefore, is usually based on the formula for torsional strength, 



AXLES, SHAFTS, AND SHAFT COUPLINGS 289 

modified by practical coefficients which experience has shown 
will provide for stiffness against torsion and bending. 
Referring to equation E, Article 12, 



* l6 T j 3 l 16 T 
dr = or a 



3 li6T ,, 



If P be the equivalent force applied at the periphery of the shaft, 
so that T = Pr, where r is the radius of the shaft in inches; and 
if N be the number of revolutions of the shaft per minute; then 
the horse power transmitted will be 

TX ■ 2PwN 2TnN 

H.P. = 



orf = 



33,000 X 12 33,000 X 12 

33,000 X 12 X H. P. 63,024 H.P. 



27tN N 

Substituting this value of T in (1) above, 






12) 



where k is a constant depending on the stress assigned. If shear- 
ing stress alone were to be considered, p s might be taken as high 
as 9,000 lbs. per square inch, for steel shafting. In order to 
secure stiffness, and to provide for the indeterminate bending in 
line shafts, it is customary to assume a lower stress (or higher 
factor of safety), depending on the material used, and the service 
for which the shaft is intended. The larger and more important 
the shaft, the lower should be the working stress, as the failure 
of a head shaft or shaft of a prime mover is accompanied by great 
inconvenience and expense. The following factors of safet} f are 
Indicated by successful practice : — 

For head shafts 15 

" line shafts carrying pulleys . . .10 
" small short shafts, countershafts, etc. 7 

For steel shafting, the allowable stress for the above factors would 
be about 4,000, 6,000, and 8,500 respectively, whence 
»9 



290 MACHINE DESIGN 

For head shafts, 



N 
For line shafting carrying pulleys, 

<* = 3-75^!^f- (4) 

For small short shafts, countershafts, etc., 

d = 3 . 3 6^Eil: ( 5 ) 

It must, however, be borne in mind that a universal rule 
cannot be laid down for any class of shafting; and cases will 
always arise which need further consideration than given by the 
above equations. For example, in the span of shafting where 
the power is applied by a large belt the bending action may be 
excessive, and this particular span may have to be of larger diam- 
eter than the remainder of the shaft. The student is referred 
to handbooks* for tabulated data on the size of transmission 
shafts for various purposes. It is to be especially noted that a 
shaft carrying a transverse load, which applies a bending moment 
to the shaft, is subjected to a reversed stress as the shaft rotates. 
If, in addition, the twisting moment varies in magnitude, the 
factor of safety, owing to complete or partial reversal of stress 
(see Arts. 25 and 26), must be high, and this accounts for the 
low stresses allowable with such shafts. 

122. Shafts Subjected to Torsion and Bending. In engine 
shafts, head shafts driven by heavy belts, and many others, the 
torsional stress is not predominant and may, in fact, be less than 
that due to bending. A full discussion of the relations which 
exist in this case has been given in Article 16 and it remains to 
show the application of this discussion to actual .cases of design. 

From Article 16 (equations K, K x and Fig. 9), it appears 
that if the bending and twisting moments can be determined for 
any section, the theoretical diameter of the shaft at that section 

* See Kent's " Mechanical Engineer's Pocket Book," page 869. 



AXLES, SHAFTS, AND SHAFT COUPLINGS 2gi 

can be found. Usually the twisting moment can be determined 
without difficulty, but the bending moment is often difficult to 
determine, and sometimes the designer must be content with an 
approximation. One of the greatest sources of uncertainty is 
the location of the reactions at the bearings. Usually, as already 
pointed out, the safe assumption is made that these reactions are 
concentrated at the centre line of the bearing. When the shaft 
is of appreciable length (15 or 20 diameters), the error is small; 
but in such cases as the crank shafts of multiple-cylinder engines, 
where the distance between the centres of bearings is only four 
or five diameters, or less, it is evident that the assumption is 
in the direction of excessive safety. 

In line shafting, particularly with the usual swivel bearings, 
the error from this source is small, and at first sight the conditions 
of such shafting would appear to approximate those of a con- 
tinuous beam. While such an assumption might be safely made 
when the shafting has been put in perfect alignment, it would not 
be safe as a general principle, as perfect alignment, even under 
best conditions, is of short duration, and bending stresses soon 
appear as a result of lack thereof. It appears, therefore, that, 
in this case, the safest procedure would be to treat each span 
as if disconnected at the bearing, when computing bending 
moments. 

A typical case of combined twisting and bending is the engine 
shaft shown in Fig. 106 (a), the data taken being those of the 
example in Case (c), Art. 5. Here the shaft is supported by the 
bearings at the points X and X f , as indicated, and carries a heavy 
generator spider at Y. The weight of this spider, and that of 
the shaft itself, with the probable magnetic pull which may occur 
when the shaft wears downward a little, is estimated at 22,000 
lbs. The maximum pressure (P) on the crank pin, due to the 
steam pressure, is 25,000 lbs. This force is a maximum when 
the crank is about vertical, and, at that position, it exerts a twist- 
ing moment on the shaft from the' crank to the point F* where 
power is delivered, and also a bending moment on the shaft in a 

* The reinforcing effect of the hub of the spider is neglected. 



292 



MACHINE DESIGN 



horizontal direction. The weight of the generator, etc., exerts 
a simple bending moment in a downward direction, and at right 




angles to that induced by P. Fig. 106 (b) shows, isometric ally, 
the direction and point of application of the various forces and 



AXLES, SHAFTS, AND SHAFT COUPLINGS 293 

reactions, and it is required to find the maximum equivalent 
bending moment on the shaft. 

It was shown in Example (c), Article 99, how a tentative solu- 
tion could be made for the diameter and length of the main jour- 
nal, thus fixing the distance of its centre line from the centre line 
of the crank pin at 2\%". Other data fix the distance between 
bearings as f — 9". 

Graphical Analysis is here very convenient, and the order of 
procedure will be as follows : — 

(a) Find the bending moment due to the steam pressure P. 

(b) Find the bending moment due to the dead load W. 

(c) Combine these bending moments to find the maximum 
resultant bending moment. 

(a) Consider, for convenience, that the force P, and all the 
reactions due to it, have been rotated into the plane of the paper 
so that P is represented as acting vertically. Draw the force * 
diagram, O' B' A' C for force P, and the reactions due to it, to 
a convenient scale, here taken as 8,000 lbs. per inch, taking O' 
on a horizontal line through A', thus making the closing string 
of the space diagram also horizontal, which is convenient for 
later work. Draw the space diagram, MNP for force P, and 
the reactions due to it, as shown. The scale of the space dia- 
gram is $4" to 1 ft. or -^ size. 

(b) In a similar manner construct the force diagram ABC O, 
for force W, and the corresponding space diagram H T J I, 
for the force W> making the pole distance = A f 0'\ taken here 
as 3".f 

(c) To combine the bending moments at any section, as Z, 
take the intercept ST, on H I J, and lay it off as S' T f on 
the diagram MNP. The distance T' U is proportional 
to the combined bending moments and may be used as an 
ordinate VV in the diagram of combined bending moments 
DGFE. m 

It often occurs that the shaft carries a heavy flywheel at F, 

* See also Article 120. 

f Reduced to one-half size in cut. 



294 MACHINE DESIGN 

instead of a generator, and a heavy belt may also run on the 
wheel. It is evident that the resultant force due to the weight 
of the wheel and the pull of the belt, can be determined, both in 
magnitude and direction. In general, the direction of this force 
will not be vertical, but will make an angle, <£, less than 90 with 
the direction of the force P. In such a case the moments may be 
combined by the triangle of forces taking into consideration the 
angle <f>. 

The numerical value of any moment is the continued product 
of the ordinate which represents it, the pole distance, the re- 
ciprocal of the scale of the space diagram, and the reciprocal of 
the scale of the force diagram. Thus the maximum bending 
moment, which occurs at 

V_ T 5//VoVl6 ..8,000 



16 



X 3 X y X = 485,400 inch pounds. 



The twisting moment is seen by inspection to be uniform 
over the whole length of the shaft which it affects. Its numerical 
value is, as before, 25,000 X 18 = 450,000 inch lbs.; and these 
two moments may be combined, to determine the safe diameter 
of the main part of the shaft according to the methods of Arti- 
cle 16. A graphical method will be given later, which some- 
what facilitates the numerical work of this computation. 

The methods outlined above are clearly applicable to any 
shaft which has not more than two points of support since in such 
cases the reactions can be readily found. 

A convenient diagram is shown in Fig. 107 for determining 
the diameter of a shaft, of solid circular cross-section, subjected 
to any moment, and with any intensity of fibre stress from zero 
to 15,000 lbs. per sq. inch. This diagram can be used for 
either simple bending or twisting moments, or for combined 
bending and twisting actions. Its use in connection with prob- 
lems involving simple twisting moments will be discussed first. 

If T is me twisting moment, d the diameter of the solid 
circular shaft, and p the intensity of stress in the most strained 

fibres, T = -— p d 3 . Therefore, for a given diameter of shaft, T 



Scale B 
6 7 8 9 




4 
Scale C 



Fig. 107. 



296 



MACHINE DESIGN 



is directly proportional to p. Thus, if d = 4", d z = 64, and 
T = .196 X 64^ = 12.57^. If p be taken as 10,000, T — 125,700 
inch lbs. In Fig. 107 if ordinates represent moments (to the 
scale " A t " of 500 inch lbs. to each division); and if abscissas 
represent intensity of stress (to the upper scale, "B," of 1,000 lbs. 
per sq. inch to each division), the point a corresponds to T = 
125,700, p = 10,000, d = 4". As the moment varies directly 
as the intensity of stress, for any given diameter of shaft, the 
relations between corresponding values of T and p (for a 4" 
shaft) will be represented by the straight line through the point 
a, and the origin O. In a similar manner straight lines through 
the origin are drawn for other shaft diameters. 

To determine the diameter of shaft for a moment of 90,000 
inch lbs., with a fibre stress of 12,000 lbs. per sq. inch, pass along 
the horizontal through the point marked "9" (or T = 90,000) 
on scale ".4," to the vertical line through the point marked "12" 
(or p = 12,000) on scale "B" The intersection of this horizon- 
tal and vertical (b) lies a little below the diagonal marked 3.4 
at its outer end; or the shaft should be about 3.3 7" (or 3H") 
diameter to give a stress of 12,000 lbs. per sq. inch. 

The oblique line nearest to the point located in the last 
example bears three figures, viz.: ''.732 — 1.58 — 3.4," and the 
other diagonals each bear three separate figures. The signifi- 
cance of these designations will be explained by further illustra- 
tions. 

If T = T V of 90,000, or 9,000 inch lbs., and p = 12,000, 

, , ' % . _J i°^ = 3 . 37 ± </75 = 1.56"; 



^ it p y 12,000 7t y 



10 



since d varies as the cube root of T, and when T = 90,000, d = 

3-37"- 

In a similar way, if T = 900, or T J ^ of 90,000, d = 3.37 -4- 

\/~ioo = -7 2 6". 

To use the diagram when T =-= 900, and p = 12,000, consider 
scale "A" as representing the moment in 100 inch lbs.; pass 
along the horizontal through 9 on this scale to the vertical, through 
12 of scale "B" as before, to the point "b" and take the first 



AXLES, SHAFTS, AND SHAFT COUPLINGS 297 

figure borne by the nearest diagonal (.732) as the approximate 
diameter of the shaft; or, by interpolation, find the diameter = 
.7 26". 

If T = 9,000, p = 12,000; consider scale "A" as represent- 
ing the moment in 1,000 inch lbs., and read the middle figure on 
the nearest diagonal (1.58) as the required approximate diameter 
of the shaft; or, by interpolation, the diameter is found to be 

1.56". 

If the moment is greater than 130,000, the diagram is quite 
as applicable as for smaller moments. Thus if T = 900,000 
and p = 12,000, consider scale "A y " as representing the 
moment in 100,000 inch lbs. The horizontal through 9 of 
scale" A" and the vertical through 12 of scale "B" intersect 
at "6" as before. The required diameter is about 7.26"; be- 
cause the diameter was found to be about .726 for a moment 
of 900, and it must be 10 times as great for a moment of io 3 X 
900 = 900,000. For p = 12,000 with a moment of 9,000,000 
inch lbs. (= io 3 X 9,000), the diameter is 10 X 1.57 = 15.7", 
etc. It thus appears that the diagram covers all moments, with- 
out being of such impracticable size as it would be if it were not 
for the peculiar designation of the oblique lines and the method 
of using scale "A" The diagram can also be used for simple 
bending moments. The expression for the bending moment in 
a shaft of solid circular section is 

M = ^pd*', 
3 2 

while the expression for a twisting moment is, as given above, 

Therefore, with a given diameter and numerically equal fibre 
stress, T is numerically equal to 2M. To determine d for given 
values of p and M, multiply M by 2 to get the equivalent T, and 
with this value of T, proceed as in the former examples. 

For finding the diameter appropriate to a combined bending 
and twisting moment, the equivalent twisting moment, 

T = M + V M 2 + T 



298 MACHINE DESIGN 

is to be determined ; see Art. 1 6, equation K 4 . This equivalent twist- 
ing moment is readily determined from the diagram by the use of 
scale "C" at the bottom of Fig. 107 and a pair of dividers, when 
the simple bending moment (M) and the simple twisting moment 
(T) are given. Example: Suppose M = 30,000; T = 40,000; 
and p = 13,000. Consider scales "A" and "C" to measure 
moments in 10,000 inch lbs. Take M at 3 on scale " A" with 
one point of the dividers, and T at 4 on scale "C" with the other 
point of the dividers; then the distance between 3 on scale "A" 
and 4 on scale "C" represents \/ M 2 + T 2 . Swing the dividers 
about the point at 3 on scale "A" as a centre until the other 
point reaches scale "A " (at point 8) ; then o. .8 on scale "A " = 

o s + 3 8 = M + V M 2 + T 2 = T e . With the value of 

T e , found in this way, proceed as in case of a simple twisting 
moment. The intersection of the horizontal through 8 (T e ) 
and the vertical through 13 (p) is at point "c." Since the 
moments correspond to units of 10,000 inch lbs. on scale " A" 
the largest figures of the diagonals are to be read in determining 
the diameter. The point "c" therefore indicates a diameter of 
between 3.0" and 3.2"; by interpolation the diameter is taken 
as 3. 1 5". By computation the diameter is found to be 3.14". 
A shaft 3 T y diameter would be proper for this case. The use 
of the diagram in connection with equations K and K 5 of Table 
VI is obvious from the above. 

The diagram of Fig. 107 is equally convenient for finding 
the intensity of stress in a given shaft under a known moment; 
or the moment on a given shaft corresponding to any intensity of 
stress. Thus, if a i%" shaft is subjected to moment of 1,000,000 
inch lbs., consider the moment units as 100,000 inch lbs., pass 
horizontally from 10 on scale u A" to a point slightly below the 
diagonal marked .776 (7.76" diameter), and then vertically up- 
ward to scale "J3," where the stress is read as about 11,000 lbs. 
per sq. inch. 

If it is required to find the twisting moment corresponding 
to an intensity of stress of 9,000 lbs. per sq. inch on a shaft i}4" 
diameter; pass vertically downward from "9" on scale "B" to 
a point slightly above the diagonal marked "1.49"; then 



AXLES, SHAFTS, AND SHAFT COUPLINGS 299 

horizontally to 5.9 on scale "A." As 1.49 is the middle number 
on the diagonal, the moment units are 1,000 inch lbs.; therefore 
T = 5.9 X 1,000 = 5,900 inch lbs. 

123. Torsional Stiffness and Deflection of Shafting. When a 
shaft has considerable length, the matter of torsional stiffness 
is important. A rule, common in practice, is to limit the twist 
in the shaft to one degree for every 20 diameters in length. 
Another rule limits the twisting to 0.075 degree for every foot 
in length. The lateral deflection of the shaft should not exceed 
yxto" per foot of length, to insure proper contact at the bearings. 
Theoretical considerations, however, do not enter so largely into 
the spacing of bearings of line shafting, as does the construction 
of the framework to which the bearings are fastened. Care 
should be exercised in laying out such structures, that provision 
is made for fastening the hangers close enough together to avoid 
excessive deflection. For the average range of velocities found 
in practice the following formulae* can be used for ordinary 
small shafting. 

L = 7 \/ d 2 for shaft without pulleys (1) 

L — 5 v'd 2 f° r shaft carrying pulleys ... (2) 

where L = distance between hangers in feet and d = diameter 
of shaft in inches. 

If T be the twisting moment in foot lbs. applied to a shaft, 
then the power transmitted at AT" revolutions per minute \s2TnN; 
from which it appears that the greater the velocity of the shaft, 
the smaller is the required turning moment, for a given amount 
of power transmitted. 

If a slightly deflected shaft is rotated, centrifugal force, acting 
on the eccentric mass of the shaft, tends to equalize the forces 
which hold the shaft deflected in one plane and to whirl the shaft 
as a whole around the axis of rotation. At low speeds the action 
of centrifugal force is small, and the deflecting force will hold the 
shaft deflected in its plane. As the effect of centrifugal force 
increases with the velocity, while the effect of the deflecting force 
is constant, it is clear that as the speed is increased the centri- 

^ ■ — * 

* See also Kent's "Mechanical Engineer's Pocket Book," page 869. 



300 MACHINE DESIGN 

fugal force will, at some speed, balance the effect of the de- 
flecting force, and the shaft will become unstable. Beyond this 
speed the shaft will whirl about the central axis. For a given 
diameter of shaft there is one definite speed within which it will 
maintain a stable condition with a given deflection. 

If L = distance between bearings in feet, d = diameter of 
shaft in inches, and N = the revolutions per minute, then for the 
critical speed * 

£ = I "# ® 

This equation refers to the bare shaft only and it determines 
the maximum safe span. Where pulleys are carried at some 
distance from the bearings, the span, L, must be less than the 
value given by equation (3) on account of the added mass of the 
pulleys, and the great liability of the latter to be unbalanced. 
The speed of shafting in practice is, almost always, considerably 
below the critical speed. 

124. Practical Considerations, Hollow Shafting, etc. Shafting 
up to 3" in diameter is, in this country, made of cold-rolled steel. 
Such shafting is true and straight and needs no turning whatever. 
If keyways are cut the shaft must, in general, be carefully straight- 
ened afterward, as the cutting relieves, locally, the skin tension 
due to the cold-rolling f thus causing the shaft to warp. Larger 
sizes of shafting are forged and machined. 

The use of hollow shafts not only reduces the weight for a 
given strength, but the removal of the metal from the core of a 
steel shaft (or of the ingot from which it is made) very greatly 
increases its reliability under repeated application of stress. 

Shortly after a steel ingot is cast, the exterior solidifies and 
becomes comparatively cool while the internal portion is still 
fluid. The subsequent contraction, during complete cooling, is 
much less in the exterior walls than it is in the hotter interior 
mass. Unless the interior is "fed" during this period, it will be 
less dense than the outer portions and shrinkage cavities are apt to 

* See Rankine's "Millwork," page 549. 
t See Article 12. 



AXLES, SHAFTS, AND SHAFT COUPLINGS 301 

be present near the centre of the ingot. Numerous expedients 
have been adopted to reduce this evil, among which is "fluid 
compression," or subjecting the ingot to heavy pressure immedi- 
ately after it is poured. The difficulty is not entirely overcome 
by such means, however, as the walls of large ingots become too 
rigid to yield to the pressure before the interior is entirely solidified. 
The external walls "freeze," after which the internal shrinkage is 
made up by metal flowing from the upper portion toward the 
bottom as long as any of it remains fluid. This leaves a shrinkage 
cavity at the upper end of the ingot. Gas liberated during cooling 
collects in this cavity also. The result of these two actions is 
to form what is called the "pipe," which frequently extends to a 
considerable depth. The top end of the ingot is cut off and 
remelted, but this does not insure removal of all of the pipe, and 
it also involves much expense. If the portion cut off is not 
sufficient to remove all of the pipe, a piece rolled or forged from 
the ingot contains a flaw near the centre which is drawn out 
into a long crack if the ingot is worked into a long piece. The 
rolling and forging may squeeze the sides of the cavity together 
so that it is not easily detected at any section, but as this work 
is done at a temperature much below that corresponding to 
welding, the defect is not removed. This flaw is more or less 
irregular or ragged; hence its form is favorable to starting a 
fracture, under variations of stress, which may finally extend 
far enough to cause rupture. 

If the ingot is bored out, the pipe is effectually removed, and 
the metal remaining is superior to that of a solid shaft. It will 
be evident that casting a hollow ingot is not the equivalent of 
boring out one which was cast solid; for if the ingot is cast hollow 
the outer and inner walls cool before the intermediate mass does, 
and the. shrinkage effect takes place in the latter. In fact, a 
shaft made from a hollow ingot is worse than the solid shaft, in 
the respect that the former has the defective material nearer the 
outer fibres where the stress is greater. 



302 



MACHINE DESIGN 



COUPLINGS AND CLUTCHES 

125. General Description. Couplings are machine members 
which fasten together the ends of two shafts, so that rotary motion 
of one causes rotary motion of the other. Where the connection 
is to be broken only at rare intervals, as in making of repairs, the 
couplings are generally constructed so that they must be partially 
or wholly dismantled to separate the shafts. Such couplings are 
known as permanent couplings. When it is desired to disengage 
the shafts at will, the coupling is of a different construction and 
is generally known as a clutch.* The use of clutches is not, how- 





Fig. 108. 



Fig. 109. 



ever, confined to securing together the ends of shafting, but they 
are much used for engaging and disengaging pulleys at will, in 
connection with the shafts on which they are placed. For this 
service clutches making use of friction are much used, and this 
particular type is discussed in Chapter XIII. 

Couplings should be placed near a bearing, so as to bring 
the joint in the shaft near a supported point, and should be placed 
on the side of the bearing farthest away from the point where 
power is applied, so that when the shaft is disconnected the 
running part is supported near the end. 

126. Permanent Couplings. Where the axes of the two shafts 
to be connected are parallel and coincident, couplings such as are 
shown in Figures 108, 109, no, and in are used. Fig. 108 
illustrates a type of coupling known as a split-muff coupling. 



* See Transactions A. S. M. E., 1908, for a full description and discussion of 
various forms of clutches. 



AXLES, SHAFTS, AND SHAFT COUPLINGS 



303 



The parts A and B are separated by a small space and can, there- 
fore, be clamped to the shaft by the bolts C. For heavy work a 
key as shown is provided, but in lighter shafting friction alone 
may suffice to prevent relative rotation. 

Fig. no shows the Sellers Muff Coupling. Here the circular 
tapered wedges B, B, are drawn inward by the bolts C. The 
wedges are split as shown at D, hence the tighter they are drawn 
inward the more firmly they clasp the shaft. For light work no 
key is necessary, but for the full capacity of the shaft keys are 
advisable. 

Couplings such as shown in Figures 108 and no are regularly 





Fig. 1 10. 



Fig. hi 



manufactured in standard sizes, and the student is referred to 
the trade catalogues of manufacturers for dimensions and capa- 
cities of such couplings. 

The Flange Coupling, Fig. 109, is one of the most common 
and also one of the most effective forms of permanent couplings. 
The general proportions are usually designed empirically, but the 
bolts should be designed so that their combined resistance to a 
torsional moment, around the axis of the shaft, will be at least as 
great as the torsional strength of the shaft itself; and the bolts 
should be accurately fitted so as to distribute the load evenly 
among them. 

Let D = diameter of the shaft in inches 
d = diameter of the bolt in inches 
n = the number of bolts 
r = radius of bolt circle in inches 
p a = allowable shearing stress per square inch, for steel. 



3°4 



MACHINE DESIGN 

n d 2 



tzD* 
Then — — p 

10 



n r p a 



whence d = .5 



nr 



(1) 



D 



Good practice gives n = 3 + — , but this number may be 

modified for convenience in spacing, etc. The bolts should be 
carefully fitted to insure that each one carries its full share of the 
load. The projecting outer flange is an important feature as it 
covers the revolving bolt heads, thus protecting workmen from 
becoming entangled. For best results the flanges should be 
pressed on to the shaft and the faces trued up in place, thus 




Fig. 112. 



insuring greater accuracy of alignment. This should be done 
in all good work. 

When great strength and reliability are desired, as in marine 
work, the flange is sometimes forged solid with the shaft, as in 
Fig. in. Here the bolt holes are sometimes bored tapering, 
and reamed after the flanges are placed together, thus insuring 
a perfect fit for the bolts, and also facilitating their withdrawal. 

When the axes of the two shafts are parallel, but not coin- 
cident, or when there is danger of parallel and coincident axes 
wearing out of coincidence, Oldham's Coupling, Fig. 112, is often 
used. It consists of two heavy flanges (A and B) , each keyed fast 
to its own respective shaft, and an intermediate disc C. The disc 
has a tongue running diametrically across each face, these tongues 



AXLES, SHAFTS, AND SHAFT COUPLINGS 



305 



being placed at right angles to. each other and fitting into grooves 
cut in the flanges. With this coupling the rate of rotation of 
the driven shaft is identical with that of the driver, or, in other 
words, the angular velocity is the same. The coupling is often 
used on the propeller shafts of small power boats. 

If the axes of the two shafts, A and B, Fig. 113, intersect and 
make an angle with each other they may be coupled together 
by means of a Hooke's Coupling or Universal Joint, as it is often 
called. In this coupling each shaft is fitted with a jaw D which 
is pin-connected to an intermediate member F. The holes in 
this intermediate member for receiving the pins G are at right 
angles to each other. With this arrangement the angular velo- 
city of the driven shaft is not the same at all points of the revolu- 
tion as that of the driver.* The construction shown in Fig. 113 




Fig. 113. 



is very common, but the difference between the angular velocity 
of the driver and that of the driven shaft is less when the con- 
struction is such that the axes of the pins G intersect. The 
construction required to make the axes of the pins intersect is 
usually more complicated than that shown in Fig. 113, and hence 
in rough work the simpler design is adopted. 

If another shaft C be coupled to B so that A and C make the 
same angle with B; if also the pins G, G in B are parallel to each 
other and all three shafts lie in the same plane; then the angular 
velocity of C will be identical with that of A and vice versa. 
Empirical practice makes the diameter of the pin G equal to 
one-half the diameter of the shaft. 

128. Positive Clutches. Positive clutches are much used for 



* See "Kinematics of Machinery," J. H. Barr, page 198. 



20 



306 



MACHINE DESIGN 



starting and stopping such machines as punch presses which 
must work intermittently. They are made in so many forms 
that a description of them would be beyond the scope of this 
work. A very full description of many forms is given in the 
Transactions of the A. S. M. E., Vol. XXX, to which reference 
has already been made. Fig. 114 illustrates the most common 
form of disengaging coupling for heavy work. The part B is 
made fast to the shaft to be driven, while part A, which is com- 
pelled to rotate by the feather F, can be moved axially .along 
the driving shaft. A ring R, fitting the groove G loosely in a 
radial direction, is connected by the pins P to an operating lever 




Fig. 114. 



Fig. 115. 



which is not shown. When the part A is moved forward till the 
jaws J engage, A will drive B positively in either direction. In 
order to facilitate the engaging of the jaws they are often made 
as in Fig. 115, but in this case the driving can be in one direction 
only. The total cross-sectional area of the jaws must be such 
that they will not shear off under the load, and the area of the jaw 
faces must be sufficient to prevent crushing. 

Frequently, for light work, only one feather is used, but two 
feathers are, in general, better, both on account of the driving 
effort and for ease of operation. 

129. Flexible Couplings. Where it is desirable to have a 
small amount of flexibility in a shaft, a flexible coupling, such as 
is shown in Fig. 116, is employed. These members are much 
used for connecting rapidly revolving machines to prime movers, 
as in the case of a dynamo directly coupled to a steam engine, 
the object being to prevent undue stress, or bearing pressure, 
from lack of accurate alignment of the two shafts. In the con- 



AXLES, SHAFTS, AND SHAFT COUPLINGS 



307 



struction shown, the shafts A and B are fitted with heavy flanges, 
F, which carry pins, P. Links of leather or other elastic material 
connect pins on one flange with pins on the other, there being as 
many links as there are pins in each flange. This arrangement 





Fig. 116. 

allows for a slight angle between the axes of the shafts, or for a 
small lack of coincidence in the axes. The pins in one disc are 
sometimes placed on a smaller diameter than those on the other, 
so that in case of failure of the links the pins will not strike and 
cause breakage. 



CHAPTER XII 
BELT, ROPE, AND CHAIN TRANSMISSION 

130. General Considerations. When power is to be trans- 
mitted from one shaft to another, especially when such shafts 
are not far apart, in such a manner that the velocity ratio of the 
two must be constant, some form of toothed gearing is usually 
employed. When, however, it is not necessary that the velocity 
ratio remain constant, flexible elastic connectors are much used. 
When the distance through which power is to be transmitted is 
comparatively short (50 feet or less) , flat belts, or ropes of cotton 
or manila, are most common; while for longer distances steel 
ropes have certain advantages. For small amounts of power, 
round belts of leather are much used. Chain drives, which are 
virtually flexible connectors running on toothed wheels, have 
lately come into extended use for transmitting power over com- 
paratively short distances. They are very efficient, maintain 
positive velocity ratio between the two shafts, and can be used 
when the distance between shafts is too great for convenient 
use of gears. 

Leather belts are made by cementing, sewing, or riveting 
together strips of leather cut from oak-tanned ox-hides. Where 
only one thickness is used they are known as single leather belts; 
where two, three, or four thicknesses are needed to obtain a heavy 
belt, they are known respectively as double, triple, and quadruple 
belts. Cotton belts are made either by weaving in a loom, or are 
built up of several layers of canvas, sewed together, with a special 
composition between each fold. They are very little used in 
this country. Rubber belts are made of several layers of canvas, 
held together with, and completely covered by a rubber com- 
position. They are very effective in wet places. Belts of raw- 
hide are also used to some extent. 

308 



EELT, ROPE, AND CHAIN TRANSMISSION 309 

The ends of all belts are joined, to make them continuous, 
either by lacing or sewing, or by some kind of special fastening 
of which there are many on the market, or by making a permanent 
joint by cementing and riveting. The latter method is much 
preferable where it can be applied, as it makes the joint practically 
as strong as the rest of the belt, and gives a smooth surface which 
runs better than any joint. Other kinds of joints reduce the 
strength of the belt from 60 to 75 per cent, but inasmuch as the 
lacing can be replaced and the belt itself has its life prolonged by 
reduced load, this initial loss of efficient strength is not as waste- 
ful as it at first appears. 

131. Theoretical Consideration of Belts and Ropes. In Fig. 
117, let A represent a pulley whose centre is at O, and which is 




^L 



+S- 




Fig. 117. 

connected by a belt as shown to the pulley B, whose centre is at 
O v When no turning moment is applied to the driving pulley A, 
the tensions in the two parts of the belt are the same, except 
possibly for friction of the bearings, and is that due to the initial 
tension with which the belt is placed upon the pulleys. Let this 
total initial tension on each side of the belt be called T 3 . 

It is evident that this initial tension will cause the belt to 
exert a pressure upon the pulley, and this pressure will induce 
a frictional resistance opposing relative sliding between the belt 
and the pulley. If now a turning moment is applied to A, and 
a resisting moment to B, the pull upon the belt due to this 
frictional resistance will increase the tension in the lower part 
of the belt, and decrease the tension in the upper part. Let 
these new total tensions be called T 1 and T 2 respectively. It is 
evident that the tendency of the belt to slip around the pulley, 



3io 



MACHINE DESIGN 



owing to the difference in tension on the two parts of the belt, 
is resisted by the frictional resistance between the belt and pulley. 
The difference in tensions tends to rotate the pulley B, and 
when the turning moment (7\ — T 2 )r 1 becomes equal to the 
resisting moment applied to B, rotation will take place. 

If the difference between T x and T 2 which is necessary to 
overcome the resisting moment, is small compared to the 
frictional resistance between the pulley and belt, no slipping 
of the belt on the pulley will occur. To obtain this result 
in practice, would necessitate the use of very large belts, 
relatively, for the power transmitted. It has been found to 
be better practice to use smaller belts and allow the belt to 
slip somewhat. 

In addition to the slipping action noted above, all belts are 
subjected to what is known as creep. Referring again to Fig. 117 
consider a piece of the belt of unit length moving on to the pulley 
under a tension T v As this piece of belt, of unit length, moves 
around with the pulley from M to AT", the tension to which it is 
subjected decreases from 7\ to T 2 and the piece, owing to its 
elasticity, shrinks in length accordingly. The pulley A, there- 
fore, continually receives a greater length of belt than it delivers, 
and the velocity of the surface of the pulley is faster than that 
of the belt which moves over it. In a similar way the pulley B 
receives a lesser length of belt than it delivers, and its surface 
velocity is slower than that of the belt which moves over its 
surface. This creeping of the belt, as it moves over the pulley, 
results in some loss of power, and is unavoidable. The total loss 
of speed due to both slip and creep should not exceed 3%; that 
is, the surface speed of the driving pulley should not exceed that 
of the driven pulley by more than 3%. Good practice limits 
this value to about 2%. When the total slip approaches 20%, 
there is danger of the belt sliding off of the pulley entirely. 

Since the pulling power of a belt is proportional to the differ- 
ence between 7\ and T 2 , it is necessary to know the relation 
which exists between these quantities. 
Let t= the tension per square inch of belt section at any point 
on the pulley. 



BELT, ROPE, AND CHAIN TRANSMISSION 



311 



c = 



t t = the tension per square inch of belt section on the tight 

side in pounds. 
t 2 = the tension per square inch of belt section on the slack 

side in pounds. 
p = the maximum allowable tension per inch of width of belt 

in pounds. 
/ == effective pull of belt per square inch of cross-section 

= (h ~ Q) in pounds. 
v = the velocity of the belt in feet per second. 
w = the weight of one cubic inch of belt in pounds. 
q = the reaction of pulley 

against one linear inch of 

belt of the width con- 
sidered, in pounds. 
c = the centrifugal force of 

one cubic inch of belt in 

pounds at the given speed. 
jj. = the coefficient of friction 

between belt and pulley. 
r = the radius of the pulley 

in inches. 
a = the angle of belt contact 

in degrees. 

= the angle of belt contact in radians = .0175 a 

The centrifugal force of one cubic inch of belt will be 

12 wv 2 

hence the centrifugal force of one linear inch of 




gr 



belt having 1 square inch of cross section will be 



12 wnr 
gr 



Let the cross-sectional area of the belt be one square inch and 
consider an elemental portion of its length as shown in Fig. 118. 
It is held in equilibrium, when slipping is impending, by the 
following forces: — 

(a) The centrifugal force = c ds 

(b) The radial reaction of the pulley against the belt = q ds. 

(c) The frictional force = p. q ds. 

(d) The tensions t and t -\- dt. 



312 MACHINE DESIGN 

Resolving all forces vertically 

.do de 

qds + c ds = t sin ■ — + (t + dt) sin — . . (i) 

Here d 6 is so small that sin may be taken as equal to 

dd 

— - in radians, without appreciable error, and the product of d t 

dd 
and sin — may be neglected. 

Hence (i) may be written 

qds + cds = tdd . . . . . (2) 

12 wv 2 

but c = and ds = r d 

gr 

_ 12 wv 2 12 wv 2 

. • . cds = ds = d e = z d f or convenience. 

g^ g 

Hence from (2) q ds = t dO —z dO = {t — z) dO . . . (3) 
From equality of moments around O 

t + dt = t + f*qds 

■*' . dt = 11 qds (4) 

Substituting in (4) the value of q ds obtained from (3) 

dt = ;x{t — z) dd 



M dt r e J 

.'. I = 11 I do 

J t t—z J 



or kg. 7 — : = ^ e (5) 

ii z 

t x — z 
and common log = 0.434 11 

'l Z 0.434 M^ 0.0076 ju. a k 

= 10 =10 =10 for convenience . (6) 

'2 & 



BELT, ROPE, AND CHAIN TRANSMISSION 313 

Now / = t x — t 2 . ' . t 2 = t x — / and substituting this value of 
t 2 in (6) and reducing 

[/ + z] io k — z } 
h = " T— = -k + * • • • • (7) 

10 — 1 c 

where C = - 



k 
IO 



and 



/ = ['.-*] L^rj = t^- z ] c • • (8) 



v- 
If (8) be multiplied through by it will express the horse- 
power (h.p.) which a belt of one square inch cross-sectional area 
will transmit or, 

f V r 1 C v 

— = h.p. = [t l -z\— .... (9) 

132. Practical Coefficients. In the above equations the 
following quantities a, ft and z, must be known or assumed before 
a solution for t x or / can be made. The angle of contact, a, can 
be taken from the drawing of the drive in question, and some 
allowance should be made for the conditions of operation. Thus 
if the belt is to run in a horizontal position, with the slack side 
on top, the full theoretical value of a may be taken. If, however, 
the slack side must be on the bottom (an arrangement which 
should be avoided if possible) or if the belt is to be run in a vertical 
position, some reduction must often be made in the theoretical 
value of a to allow for sagging of the belt. This also applies to 
belts running at high speed, where centrifugal force tends to 
lessen the arc of contact. 

The coefficient of friction p. is an exceedingly variable quan- 
tity, changing with the character and the condition of the surfaces 
of contact, the initial tension of the belt, and the rate of slip. It 
has been found by experiment that, within reasonable limits, the 
coefficient increases with the slip and that, as before stated, a 
maximum rate of slip, including creep, not in excess of about 
3 per cent is good practice. Experiments made by Professor Dieder- 
richs in the laboratories of Sibley College gave the values of n 



314 



MACHINE DESIGN 



shown in the first column of the following table. Allowing for the 
difference between conditions in the laboratory and those found 
in practice, the value shown in the second column may be used 
in designing leather belts. 

For pulleys made of pulp, /^ • = 0.29 0.20 

For pulleys made of wood, fi = 0.31 0.22 

For pulleys made of cast iron, \i = o . 46 o . 30 

Values considerably above these were found for paper pulleys of 
special construction. 

The quantity z is proportional to the weight of the belt per 
cubic inch. For ordinary leather (which is most commonly 
used), w may be taken from 0.03 to 0.04, an average value being 
0.035 pounds. 

Table XVIII has been calculated with a value of w = 0.035, 
while Table XIX is abbreviated from " Transmission of Power 
by Belting"* by Wilfred Lewis. 



TABLE XVIII 



Values of z — 



12 wv' 
g 



, for v = ft. per sec, or 







V = ft 


per minute, 


w — 


°3S- 












V 


30 


40 


50 


60 


70 


80 


90 


100 


no 


120 


130 


140 


V 


1,800 


2,400 


3,000 


3,600 


4,200 


4,800 


5>4©o 


6,000 


6,600 


7,200 


7,800 


8,400 


z 


n-75 


20.9 


32-5 


47.0 


64.2 


83-4 


105 -5 


i3°-5 


157-6 


187.6 


220.2 


255-5 



Example. Design a belt to operate a dynamo of 15 H.P. 
capacity, when the belt velocity is 2,400 ft. per minute. Assume 
fi = 0.30, a = 180 and t x = 200 lbs. 

From equation (9) the horse-power transmitted by a belt 
having a cross-sectional area of one square inch is for these con- 
ditions: 

7 r n C v , .61 X 40 

h. p. = [t x — z\ = [200 — 20 . oj — ^ — = 7 . 9 



55o 



55o 



15 



.'. the cross-section required = — = 1.0 sq. in. 

7-9 

which is equivalent to a belt ^ thick and 8" wide. 

* Transactions A. S. M. E., Vol. VII, page 579. 



BELT, ROPE, AND CHAIN TRANSMISSION 



315 



The total tension (7\) in the tight side of the belt will be 
1.9 X 200 = 380 lbs. The total tension (T 2 ) in the slack side 
will be this value minus the required effective pull, P, which is 
found by dividing the foot pounds of work to be done by the 

15 X 33,000 

velocity of the belt or, P = = 206. 

2,400 

7\ — P = 380 — 206 = 174 pounds. 

TABLE XIX 



Hence T, 



Values of C = 



(Nagle) 











Degrees of Contact 


= a 








P- 


90 


100 


no 


120 


130 


140 


I50 


160 


170 


180 


• 15 


.210 


.230 


• 250 


.270 


.288 


•3°7 


•3 2 5 


•342 


•359 


•376 


.20 


.270 


•295 


•3 J 9 


•342 


•364 


.386 


.408 


.428 


•448 


.467 


• 25 


•3 2 5 


•354 


.381 


.407 


•432 


•457 


.480 


•5°3 


•5 2 4 


•544 


■3° 


•376 


.408 


.438 


.467 


•494 


.520 


•544 


•567 


•590 


.610 


•35 


•423 


•457 


.489 


.520 


•548 


•575 


.600 


.624 


.646 


.667 


.40 


.467 


.502 


•536 


•567 


•597 


.624 


.649 


•673 


•695 


•715 


•45 


•507 


•544 


•579 


.610 


.640 


.667 


.692 


•7i5 


•737 


•757 


•55 


•578 


.617 


.652 


.684 


•7 J 3 


•739 


•763 


•785 


.805 


.822 



Equations (7) and (8) involve the relations which exist between 
T l and T 2 for a given set of conditions, but they do not indicate 
the relation between them and the initial tension T 3 . It was 
formerly supposed that the sum of 7\ and T 2 was constant and 
equal to 2T 3 ; and this relation may still be used for very rough 
calculations. Mr. Wilfred Lewis* has shown, experimentally, 
that this is not true. The ratio of stress to strain in leather and 
rubber increases with the strain instead of being proportional to 
it as in ductile metals. When a belt transmits power the tension 
is increased on the tight side and decreased on the slack side till 
the difference in tension is equal to the required driving force. 

* See Transactions A. S. M. E., Vol. VII, page 566. 



31 6 MACHINE DESIGN 

This is accomplished by what virtually amounts to shortening 
the belt on the tight side, a given amount, by transferring this 
amount to the slack side. Because, however, of the relation 
between stress and strain noted above, the increase of tension on 
the tight side, due to this amount of shortening, is greater than 
the decrease of tension on the slack side due to an equal amount 
of lengthening, and, as a consequence, the sum of the two ten- 
sions is increased* as the effective pull is increased. Sugges- 
tion: Place a rubber band over the fingers of the two hands 
and stretch it moderately; then twist one of the hands in either 
direction and the increase of force tending to bring the hands 
together will be apparent. 

In the case of a long horizontal belt the increase in the sum 
of the tensions is still further augmented in driving, because 
the tension on the slack side (with a proper initial tension in the 
belt) is largely due to the sag of the belt from its own weight; 
and thus the tension on the slack side tends to remain nearly 
constant, while the tension on the tight side increases with the 
power transmitted, at a given speed. It is found that the sum 
of the tensions on the two sides, when driving, may exceed the sum 
of the initial tensions by about 33 percent in vertical belts, and in 
horizontal belts the increase may be limited only by the strength 
of the belt. In addition to the causes discussed, the tension on 
both parts of the belt are increased by the centrifugal action due 
to the mass of that portion of the belt which is rotating round 
the pulley axis. This latter cause increases the stresses on both 
the tight and slack sides of the belt, and decreases adhesion be- 
tween the belt and the pulley, but does not increase the loads on 
the shafts which produce pressure at the bearings and flexure of 
the shafts. 

Large belts should therefore be put on with care, as to initial 
tension. Ordinarily, the initial tension is left to trained judg- 
ment, but it would seem that the more advanced practice of 
splicing the belt under a known initial tension will add to the 
life of large and important belts. 

* See Transactions A. S. M. E., Vol. VII, page 569. 



BELT, ROPE, AND CHAIN TRANSMISSION 317 

133. Strength of Belting. The ultimate strength of good 
leather belting will vary from 3,500 to 6,000 pounds per square 
inch. Professor Benjamin * gives the strength of cotton belting as 
about the same as good leather. He also found that four-ply 
rubber belting had a tensile strength of from 840 to 930 pounds 
per inch of width. The ultimate strength of belting seldom 
enters as a factor in belt design, as the real strength of the belt 
is in the joint. Where the ends of the belt are laced together, a 
maximum working stress of 200 to 300 pounds per sq. inch is 
found to be good practice; and where the belt is cemented to- 
gether, thus making it " endless," a working stress of 400 pounds 
per square inch may be used. The thickness of leather belting 
varies from ^ to -3V inch for single leather, and from yi to %. 
inch for double leather. Hence for single leather, 

p = 50 to 75 pounds per inch of width for laced belts. 

p = 100 pounds per inch of width for cemented belts. 
For double leather belts p may be taken at twice these values. 
Lower stresses than these are often advocated, and undoubtedly 
lower stresses increase the life of the belt. 

134. Velocity of Belting. In equation (8) when z = t v f = o 
and the belt will exert no turning force, the centrifugal force 
relieving all frictional resistance between the belt and pulley. 

If t t be taken as high as 400 pounds, and w = .035 this will 

12 w v 2 
occur when z = 400 or when = 400 whence v = 175 ft. 

o 

per second or 10,500 feet per minute. 

If equation (8) be multiplied through by v, the velocity of 
the belt, it will express the rate at which energy is being de- 
livered, or 

, _ _ _ r 1 2 w v 2 -\ 

fv = v[t 1 -z] C = v 1^ - : — — J C 

If now fi = .3, w = .035, a = 180, which are average conditions, 
the equation becomes 

J v = v [t t — .013 v 2 ] X 0.6 = 0.6 t x v — .0078 v 3 

* See "Machine Design," by Benjamin, page 186. 



318 MACHINE DESIGN 

Differentiating the right-hand side with respect to v and equating 
to zero 

0.6 t 1 — .0234 v 2 = or v =-5:1 V/i • • ( IO ) 
which gives the relation between v and t x for maximum power. 
When t x = 400, v = 102 feet per second or 6,120 feet per minute 
and when t x = 275 pounds, v — 85 feet per second, or 5,100 feet 
per minute. It is often necessary to run belts at much lower 
speeds than these; but it is not economical to exceed these limits. 
A speed of a mile per minute may be taken as about the economi- 
cal maximum limit; and it so happens that this is also about the 
limit of safety for ordinary cast-iron pulley rims. For durability 
combined with efficiency, a speed of 3,000 to 4,000 feet per 
minute may be taken as a fair value, though practical limitations 
such as speed of shafting and diameter of pulleys often fix belt 
velocities at much lower values. 

135. Efficiency of Belting. The losses of power in belt trans- 
mission consist of the loss due to slip and creep, that due to bend- 
ing the belt over the pulley, and the frictional losses at the shaft 
bearings, due to belt pull. The first two, slip and creep, should 
not exceed 3 per cent, and 2 per cent is better. The loss due to 
bending the belt is, usually, negligible although the effect on the 
life of thick belting running on small pulleys is important. The 
losses at the bearings may be considerable if the belt must be 
laced on under great initial tension in order to carry the load, and 
this condition should be avoided except where it is absolutely nec- 
essary to use a short belt. A well-designed belt transmission 
should have an efficiency at least as high as 95 per cent, and it 
may be as high as 97 per cent including bearing losses. 

136. Other Equations, Common Rules. If in equation (9), 
w be taken as 0.032 and t x as 305 pounds the equation reduces to 

h"P- = [ -55 ~ 0.0000216 v 2 ] vC . . . (n) 

io k -i 
where C = — as before and h. p = horse-power per square 

inch of belt area. If the equation be multiplied by A , the area of 
the belt cross-section, it will express the total horse-power trans- 
mitted, or H. P. =[ .55 — 0.0000216 v 2 ] v C A .... (12) 



BELT, ROPE, AND CHAIN TRANSMISSION 319 

Professor Diederichs has pointed out that equation (12) is iden- 
tical with that reported by Mr. Nagle to the A. S. M. E.* and 
commonly known by his name. Values of C have already been 
given in Table XIX. 

In the transactions of the American Society of Mechanical 
Engineers, January, 1909, Mr. Carl Barth presents a more ex- 
tended mathematical treatment of the driving capacity of belts. 
He also presents scientific methods for measuring the tension 
in belting. Many other formulae of a strictly empirical char- 
acter are given by different authorities and some of them 
are very convenient. In general these last formulae neglect 
centrifugal action and are hence applicable only to belt speeds 
below 2,500 feet per minute. Thus a common rule is that a 
single leather belt one inch wide traveling 1,000 feet per 
minute will transmit 1 H.P. Kent's " Mechanical Engineer's 
Pocket Book," page 877, gives a number of these so-called 
practical rules. 

137. Practical Considerations. One of the most valuable 
contributions to the literature of the subject is " Notes on 
Belting," by Mr. F. M. Taylor, in Vol. XV of the Transac- 
tions of the American Society of Mechanical Engineers. Mr. 
Taylor kept an accurate record of measurements and observa- 
tions on belts in use at the Midvale Steel Co.'s works, for nine 
years, and gives many valuable facts and practical suggestions 
in his paper. A satisfactory abstract of it is not possible here. 
Mr. Taylor advocates thick narrow belts rather than thin wide 
belts.| He sums up his investigation in 36 " Conclusions," 
among which are : 

"A double leather belt having an arc of 180 will give an 
effective pull on the face of the pulley per inch of width of belt 
of 35 pounds for oak-tarmed and fulled leather, or 30 pounds for 
other types of leather belts and 6- to 7-ply rubber belts." 

"The number of lineal feet of double belting, 1 inch wide, 

* Vol. II, page 91. 

f While in general this conclusion is justifiable, care should be taken that it 
is not carried to the extreme where the life of the belt may be shortened by ex- 
cessive bending. 



320 MACHINE DESIGN 

passing around a pulley per minute, required to transmit one 
horse-power is 950 feet for oak-tanned and fulled leather belt, 
and 1,100 feet for other types of leather belts, and 6- to 7-ply 
rubber belts." 

"The most economical average total load for double belting, 
is 65 to 73 pounds per inch of width, i.e., 200 to 225 pounds per 
square inch of section. This corresponds to an effective pulling 
power of 30 pounds per inch of width." 

"The speed at which belting runs has comparatively little 
effect on its life, till it passes 2,500 or 3,000 feet per minute." 

"The belt speed for maximum economy* should be from 4,000 
to 4,500 feet per minute." 

It should be especially noted that Mr. Taylor advocates a 
maximum belt tension of about one-half that ordinarily used. 
This would, of course, increase the first cost of the installation 
materially. His values, however, are not based on the minimum 
size of belt required to simply transmit a given horse-power, but 
on the size of belt which will transmit that horse-power for a given 
time with minimum wear and loss of time due to breakage or 
taking up to restore tension. Whether his practice is followed 
or not, it indicates the true aspect of the problem, and is a step 
in advance. 

In laying out belt drives, care should be taken to keep the 
diameters of pulleys reasonably large. The constant bending 
action to which the belt is subjected as it runs around the pulley 
is a great source of wear, and where the pulley is very small, 
compared to the thickness of the belt, this may be excessive. 
For this reason also it is probably better to run the hair side of 
the belt next to the face of the pulley as this side is more easily 
cracked by bending, than the flesh side, which is more soft 
and pliable. Mr. Taylor says it is safe to run double leather 
belts on pulleys 12 inches in diameter. 

The total length of the belt or distance between shaft centres 
also deserves attention. A belt being elastic, acts like a spring 
when tension is applied to it. The longer the belt the greater 
will be the total stretch for a given load. Suddenly applied loads, 
therefore, produce less stress in long belts than in short ones 



BELT, ROPE, AND CHAIN TRANSMISSION 32 1 

(see Art. 24). If, however, the distance between centres is too 
great, compared to the size of the belt, the belt is liable to flap 
and run unevenly on the pulleys. For small, narrow belts a 
maximum distance of 15 feet is good practice, while for heavier 
belts 25 feet is found satisfactory. 

A number of important investigations of belt transmission 
have been reported to the American Society of Mechanical 
Engineers. See the following papers in the transactions of the 
Society by: Mr. A. F. Nagle, Vol. II, page 91; Professor G. 
Lanza, Vol. VII, page 347; Mr. Wilfred Lewis, Vol. VII, page 
549; Mr. F. W. Taylor, Vol. XV, page 204; Professor W. S. 
Aldrich, Vol. XX, page 136. Abstracts of these as well as other 
valuable data are given in Kent's - " Mechanical Engineer's Pocket 
Book," pages 876 to 887. 

FIBROUS ROPE DRIVES 

138. General Considerations. When the amount of power 
to be transmitted is large, the width of belt required may be 
excessive, even when the belt is made very thick. To run wide 
belts successfully, the shafting must be kept in perfect parallel 
alignment, and the distance between shaft centres must not be 
too great. For these reasons rope drives have been found very 
satisfactory where the amount of power to be transmitted is 
large, and the distance of ' transmission relatively great. They 
are also particularly serviceable for connecting shafts which are 
not parallel, as in the case of "quarter-turn" drives, especially 
where a belt would have to be of considerable width and would, 
as a consequence, run badly. 

In all fibrous rope drives the surfaces of the pulleys or "sheaves " 
are provided with wedge-shaped grooves to receive the rope and 
thereby give the rope a better grip on the sheave. For drives 
of moderate length, 40 to 150 feet, fibrous ropes of cotton, hemp 
or manila fibre are chiefly employed. For transmitting power 
comparatively great distances, wire rope is more common, al- 
though fibrous ropes are also used for comparatively long trans- 
missions. In all long-distance transmission the rope must be 
supported at intervals by idler pulleys. 
21 



322 



MACHINE DESIGN 



Fig. 119* shows a typical rope drive where the line shafting 
of each floor of a mill is driven by its own rope drive from the 
main shaft of the engine. 

139. Materials for Fibrous Ropes. Round ropes of leather, 
or rawhide, are used to a limited extent, when the amount of 
power to be transmitted is small. Rawhide is especially useful 
in damp places, but since it costs about six times as much as 
vegetable fibre rope, its application is very limited. Leather 
belts or ropes of square | or wedge-shaped section have also been 




Fig. 119. 

used to a limited extent. In certain localities in Great Britain, 
hemp, which is a local product, is quite extensively used; but 
cotton and manila fibre are by far the most common for trans- 
missions of any considerable size. In this country manila fibre is 
used almost exclusively, while in England and on the Continent 
cotton rope is also much employed. 

It is obvious that as a twisted rope of any fibrous material 
bends while passing over the sheave, there must be a certain 



* Reproduced by permission from " The Blue Book of Rope Transmission." 
f For a fuller discussion of such ropes see " Machine Design," by H. J. Spooner. 



BELT, ROPE, AND CHAIN TRANSMISSION 323 

amount of internal friction. The result of this action is very 
noticeable in any old manila rope which has been used without 
lubrication. When such a rope is broken open it is found to be 
filled with powdered fibre, due to the internal chafing. For this 
reason manila fibre, which is naturally rough, is usually lubri- 
cated, while being twisted into rope, with tallow, paraffme, 
soapstone, graphite, or some such lubricant. 

Cotton fibres, on the other hand, are smoother and hence give 
rise to less internal friction. They are, therefore, usually laid 
up dry into rope, a dressing or lubricant being applied to the 
exterior to prevent small fibres from rising on the outside, thus 
starting the rope to fraying. This dressing also excludes mois- 
ture and retains the natural oils in the interior fibres. Cotton 
fibre is not as strong as manila. 

Professor Flather* makes the following comparison between 
cotton and manila rope: "As compared with manila, then, the ad- 
vantages of cotton ropes of the same diameter are: Greater 
flexibility, greater elasticity, less internal wear and loss of power 
due to bending of the fibres, and the use of smaller pulleys for a 
given diameter of rope. Its disadvantages are: Greater first 
cost, lesser strength, and possibly a greater loss of power due to 
pulling the ungreased rope out of the groove — in any case this 
is usually small with speeds over 2,000 feet per minute." 

140. Theoretical Considerations. The general equations 
(7), (8), and (9), of Art. 131, which were deduced for flat belts 
hold also for round ropes if the proper notation be substituted. 
In these equations the unit mass of belt was taken as one cubic 
inch. With ropes it is more convenient to take a piece of rope 
one inch in length and one inch in diameter. With the following 
exceptions, therefore, the notation used here will be the same as 
that used in Art. 131. 

Let w' = the weight of a piece of rope 1 inch in diameter and 
1 inch long. 

1 2 w' v 2 

Let z' = where w f has the value above. 

g . 

* " Rope Driving," by J. J. Flather, page 81. 



3 2 4 



MACHINE DESIGN 



Let t\ = the tension in a rope of i inch diameter on the 
tight side. 

Let C — a new coefficient = C modified on account of 
wedging effect of groove. 
Then equations (8) and (9) become 

j = \t\-z>\c> (13) 

C- v 

mdh.p. = \t\ - z'} (14) 

55o 

In equations (8) and (9) the frictional force between the 

pulley and the belt for a flat belt is taken as fi q where q is the 

radial pressure between the pulley and the belt. In a grooved 

pulley the pressure between the pulley and the rope is greater 

e 

than the radial pressure in the ratio of cosec — to unity, where 

is the angle between the sides of the groove. The frictional 

6 
resistance between the rope and sheave is therefore fi q cosec — . 

If jm cosec — be substituted for fi in the quantity C (equations 

8 and 9) the result C may be used as indicated in equations 
(13) and (14) for rope drives. The value of p. for rope sheaves 
has not been determined with any degree of accuracy. Professor 
Flather* after reviewing what experimental data there is on the 
subject, concludes that 0.12 is a fair value and computes the 

6 

following values of <f> = fi cosec — = 0.12 cosec — 

TABLE XX 

e 

(/> = coefficient of friction = 0.12 cosec — 



Angle of groove. 


3°° 


35° 


40 


45° 


5o° 


55° 


6o° 


+ 


.46 


.40 


•35 


■3 1 


.28 


.26 


.24 


It is obvious that if <j> be used instead of jut in Table XIX, the 
corresponding values of C in Table XIX will be the new constant 



* " Rope Driving,'' page 112. 



! 



BELT, ROPE, AND CHAIN TRANSMISSION 325 

C. Thus if = 45 , <j> = .31. If also a = 180 , C from Table 
XIX = .61 about. The angle 45 has been found to be the most 
satisfactory and is most commonly used. If the angle be less 
than 45 , the wedging action, hence the pulling capacity is in- 
creased, but the power loss and wear of rope due to drawing 
it out of the grooves is greater. For such sheaves, with 9 = 45 
and a = 180 

h.p~.6i[t\-z>]-^ .... (15) 

As before stated, reliable data on the coefficient of friction 
for ropes are scarce, and designing engineers have approached 
the problem of rope drives without regard to this coefficient. 
One of the most important contributions to the subject is that of 
Mr. C. W. Hunt (see Transactions A. S. M. E., Vol. XII). The 
notation of Mr. Hunt's article has been changed somewhat to 
correspond with that used in this text. 

Let d = diameter of the rope in inches. 
d = sag of rope in inches. 
L = distance between pulleys in feet. 
w f = weight of one inch of rope oj one-inch diameter. 
W = weight of one foot of rope of diameter d. 
T l = total tension in rope on tight side. 
T 2 = total tension in rope on slack side. 
T = tension necessary to give the rope adhesion. 
K = the total tension applied to each side of the rope due 

to centrifugal force. 
P = effective turning force = T 1 — T 2 

Then 7\ = T + K + P 

and T 2 = T + K 

Mr. Hunt says that "when a rope runs in a groove whose 
sides are inclined toward each other at an angle of 45 there is 
sufficient adhesion when 7\ -v- T 2 = 2. However, he assumes a 
somewhat different ratio in the development of his equation, for 
which he assumes "that the tension on the slack side necessary 
for giving adhesion is equal to one-half the force doing useful 
work on the driving side of the rope." 



326 MACHINE DESIGN 

Or T = - and T X = T + K + P = -+K + P = ^-P + K 

6 2 ° 2 2 

P 

and T 2 = T o + K = - + K by assumption. 

.-.P^-lT.-K] (16) 

If equation (16) be multiplied through by — it will express the total 

horse-power transmitted or 

H.P. =-[T 1 -K]— .... (17) 
3 55o 

The tension K on each side of the rope for an arc of contact of 

1 2 w r v 2 
180 and a rope of one inch diameter is , which is iden- 

g 
tical with the constant z' in equation (14). Mr. Hunt's formula 
therefore may be written 

h.p.=- [t\ - Z ']^- = ^L [f -z'} . . (18) 
3 55° 825 

where h.p. is the horse-power transmitted by a rope one inch in 
diameter. This is identical in form with the theoretical equation 
(15) and differs from it only by a negligible amount in the value 
of the coefficient. 

It would seem therefore that Mr. Hunt's assumptions give 
results very close to those obtained by using the value 0.12 for ^ 
as recommended by Professor Flather. 

It is to be noted that the values of z given in Table XVIII 
may be used in computing values of z' . The quantities are the 
same except for the weight w' '. In Table XVIII, w = the weight 
of one cubic inch oj leather = .035. Inequation (18), w' = the 
weight oj one inch oj rope oj one inch diameter = .028 for manila 
rope and .022 for cotton rope. If, therefore, the values given in 

Table XVIII are multiplied by - they are applicable to manila 
ropes, and if multiplied by - they may be used for cotton ropes. 



BELT, ROPE, AND CHAIN TRANSMISSION 327 

Example. What diameter of manila rope is necessary to 
transmit 25 H.P. when running 4,000 feet per minute, in grooves 
having an angle of 45 . Take t\ = 200 pounds, and w r = .028. 
From Table XVIII, z, for the given velocity = 64 nearly. . * . z' = 

64 X — = 51. From equation (18) the horse-power which a rope 

one inch in diameter will deliver under these conditions is 

_ v , 66% 

h.p. = [t\ - 25']— = [200 - $A^— = I2.I k.p. 

.*. the cross- section required = = twice the area of a one- 

12. 1 

inch rope which corresponds to a rope i¥&" in diameter. 

Fig. 120* shows curves based on equation (17), giving the total 
horse-power transmitted by ropes of various sizes for T 1 = 2ood 2 , 
and will be found convenient for making calculations. 

141. Strength of Fibrous Ropes. The ultimate strength of 
manila transmission ropes may be taken as about y,oood 2 and 
for cotton rope as about 4,6ood 2 where d = diameter of rope in 
inches. The working stress must be taken very much less than 
these values or otherwise the life of the rope is much shortened. 
For manila rope Mr. Hunt recommends that the working tension 
(7\) be not over 200 d 2 . The same factor of safety would give 
130 d 2 as the allowable working tension for cotton ropes; but 
since cotton ropes are somewhat less affected by internal chafing 
the working tension may, perhaps, be safely taken at a rather 
higher value. 

142. Velocity of Fibrous Ropes. The centrifugal force 



1 2 w' v 2 



produces a tension in a rope of one inch diameter of z f = 

1 2 w' d 2 v 2 
or in a rope of diameter d the centrifugal force = . The 

o 

allowable stress in the rope is 200 d 2 . The centrifugal force will 

12 w' d 2 v 2 

equal the allowable tensile stress when = 200 d 2 or 

g 

* From "The Blue Book of Rope Transmission," by the American Mfg. Co. 



328 



MACHINE DESIGN 



when v = 140 feet per second, at which speed the effective pull 
becomes zero for this allowable working stress. 

If equation (18) be differentiated and the differential be equated 
to zero as in Art. 134, the resultant equation will give the value 



55 



45 



40 



35 



30 



25 



20 



15 



10 



















2" 


Rope 






































































































































































1? 


i"R 


ope 


















































































































+3 


GO 

a 
















1* 


>"R 


)pe 












S 

GO 


































as 
u 


u 







































































GO 

u 

















l*i 


i ~R 


ope 












GO 



-w- 


W 
















u 


^R 


ppe~ 


























































l"Rc 


>pe 






























%"B 


ape 






























M" Rope 






























































































1 


?eet 


per 


Miri 


ute 
















V 


■i 


C> 


J 


e» 


3 


1 


> 
> 

• 


if 


3 


c 


3 
5 


P 




i 

1 


3 8 
5 3 



55 



50 



45 



40 



35 



30 



25 



20 



15 



10 



Fig. 120. 



of the velocity where the work done is a maximum, for a rope 
of one inch in diameter. This is found to be about 4,900 feet 
per minute. Since the centrifugal force, and the total working 
stress, both vary as the area of the rope this limiting velocity 



BELT, ROPE, AND CHAIN TRANSMISSION 



329 



applies to all sizes of ropes, a conclusion which is borne out by 
the curves of Fig. 120. 

It has been found, in practice, that the most economical speed 
for ropes is from 4,000 to 5,000 feet per minute. If speeds 
greater than this are used, the wear on the rope is excessive. 
For a fixed value of 7\ = 200 d 2 the first cost of a rope is a mini- 
mum at about 4,900 feet as above, and this first cost is greater 
by 10 per cent if the velocity is increased to 6,000, or decreased to 
3,700 feet per minute. The first cost is increased 50 per cent 
when the velocity is reduced to 2,400 feet per minute with 




Fig. 121. 



T t = 200 d 2 but the reduction in speed increases the life of the 
rope. 

143. Systems of Rope-Driving. There are two methods of 
placing fibrous ropes on the sheaves. In the Multiple, or English 
system, several separate ropes run side by side, each rope forming 
a closed circuit in exactly the same manner as a flat belt, and 
running constantly in its own particular groove on each pulley. 
In the Continuous or American system one rope only is used, the 
rope being carried continuously from one pulley to the other till 
all the grooves are filled, and it is then spliced ; so that the rope 
as it leaves the last groove of the driven sheave is returned 
to the first groove of the driver, or driving pulley, by means 
of an idler, or guiding sheave. This idler is usually arranged 
so that through it a suitable tension may be put upon the rope 
(see Fig. 121). 



330 MACHINE DESIGN 

Regarding the merits of the two systems it may be said that 
the multiple system is the simpler, and that it also provides 
considerable security against the loss of time due to breakdowns, 
as it is not likely that more than one rope will break at a time. 
When failure of a rope does occur, the broken rope may be 
removed and repaired at a more convenient opportunity, allowing 
the other ropes to carry the load temporarily. Occasionally, 
however, the breaking of a rope in the multiple system may cause 
great delay, on account of the broken rope becoming entangled 
in one of the rope sheaves and winding up upon it before the 
machinery can be stopped. In this system the individual ropes 
must be respliced occasionally to take up the sag in the rope due 
to stretching. The velocity ratio transmitted by a new rope will 
be different from that transmitted by an old one which has worn 
smaller, and hence fits down farther into the grooves, thereby 
changing its effective radius. The velocity ratio of the two 
sheaves can, however, have but one value, and, therefore, the 
tendency will be for either the old or the new ropes to carry the 
whoie load. When the driving sheave is the larger, this will 
result in a tendency to throw more load on the old ropes; when 
the driving sheave is the smaller the tendency is to throw more 
load on the larger and new ropes. The unequal speed of the 
ropes, of course, leads to unequal stress; and slipping and con- 
sequent wear are sure to occur. 

The continuous system is more flexible in its application than 
the multiple system; for, owing to the limited sag in the ropes 
due to the action of the weighted idler, the rope may be run 
safely at any angle. This form of drive is, therefore, much 
used for vertical and quarter-turn drives, and, generally, where 
the transmission is of a complicated nature. The principal 
objections to the system are the danger of loss of time due to a 
breakdown, and the unequal straining of the various spans of 
the rope particularly with a varying load or inequality of grooves. 
When a load is suddenly applied to the continuous system all 
the spans on the slack side become slacker except that which 
runs over the idler and which is kept at a fixed tension. A much 
greater load is hence brought on the driving span of rope next to 



BELT, ROPE, AND CHAIN TRANSMISSION 



331 



the idler and some time must elapse before this load can be 
equalized over all the spans. Mr. T. Spencer Miller * has pointed 
out that the general tendency to unequal straining may be some- 
what obviated, where the sheaves are of different diameters, by 
making the angle of the groove in the small sheave somewhat 
sharper than that in the larger, so that the product of the arc of 
contact and the cosecant of half the groove angle are equal; 
thus making the tendency to slip equal. 

The above are the principal points of difference between the 
two systems. The particular conditions of the installation must 
be considered in making a choice between them. 

144. Sheaves for Fibrous Ropes. The sheaves over which 
ropes are to run deserve special attention. Care should be taken 




Fig. 122 (a). 



Fig. 122 (b). 



Fig. 122 (c). 



that the form of the grooves, and, the effective diameters are the 
same for all grooves of the same sheave and the surfaces should 
be accurately finished and well polished, as any roughness or 
unevenness seriously affects the life of the rope. As the result 
of much experimentation two forms of grooves as shown in Fig. 
122 (a) and 122 (b) have become most common. In Fig. 122 (b) 
the sides of the groove are straight while in 122 (a) the sides are 
curved. This curving of the sides makes the angle of the groove 
somewhat flatter at the bottom and hence when the rope has been 
reduced in diameter from wear it lies lower in the groove and will 
slip a little more readily than when it is new and occupies a higher 
position. This is of importance in relieving the old rope of a 
tendency to pull harder as indicated in the preceding article. 
The curved outline is also said to assist the rope to roll in the 



* Transactions A. S. M. E. Vol. XII, page 243. 



332 MACHINE DESIGN 

groove, a very desirable feature since it distributes the wear on 
the rope. The curved groove is therefore much used in the 
multiple system. In the continuous system the rope necessarily 
rotates as it passes round the idler to the first groove. 

The angle of the groove, as before stated, is usually 45 . 
The grooves of idler pulleys for simply supporting the rope when 
the stretch is great are not made v-shaped but as shown in Fig. 
122 (c). 

The wear of fibrous ropes is both internal and external, the 
internal wear being due largely to chafing of the fibres on 
each other in bending the rope over the sheaves. For this rea- 
son sheaves should be as large as possible, and, in general, 
should not have a diameter less than forty diameters of the rope. 

145. Deflection or Sag. Where the span between the pulleys 
is considerable the amount of deflection is sometimes of import- 
ance. Since the deflection varies with the distance between 
pulleys, the size and speed of the rope and the difference in 
elevation of the pulleys, it is impossible to express the relation 
existing between them in a single formula. For the simple case 
of the horizontal drive the approximate deflection on the driving 
side may be determined both for the continuous and multiple 
systems and also the deflection of the slack side of the continuous 
system, where uniform tension is maintained by a tension weight. 
In the multiple system, however, ample allowance must be made 
on the slack side, as new ropes stretch very rapidly, and the 
deflection may become excessive before resplicing can be per- 
formed. Mr. Hunt gives the following equation (transformed), 
for computing the deflection in horizontal drives: 

A = JL-JTm. . . . (I9) 

2 W >/ 4 W 2 8 

Where T is the total tension on either the slack or tight side de- 
pending on the side for which it is desired to compute the deflec- 
tion, W the weight of rope per foot, L the span in feet and A the 
deflection in feet. Where the tension on the driving side is 
assumed to be equal to 200 d 2 , regardless of speed, the deflection 
on the driving side will be constant for a given span. As the 



BELT, ROPE, AND CHAIN TRANSMISSION 



333 



tension in the rope due to centrifugal action increases as the square 
of the velocity, there is an increasing total tension T 2 on the slack 
side for a fixed value of 7\ ; and hence the deflection on the slack 
side decreases with the velocity, the span remaining constant. 
The value of T 2 may be computed and substituted in equation 
(19) to find the deflection. 

Mr. Frederick Green * gives the following approximate 
formula for computing the deflection : 



A 



W XL 2 
ST 



(20) 



Where the symbols are the same as in equation (19), and from 
which he has calculated the following table on the assumption 
that T 1 = 200 d\ 



TABLE XXI 



Distance 

between 

Pulleys, 

Feet. 


Sag on 

Driving Side, 

All Speeds, 

Feet. 


Sag on Slack Side. 


Velocity, Feet per Minute. 




3,000 


4,000 


4.500 


5, 000 


5.500 


3° 
40 

50 

60 

70 

80 

90 

100 

120 

140 

160 


.19 

•34 

•53 
.76 

1 .0 

1.4 

i-7 
2 .1 

3-o 
4.1 

5-4 


•45 
.80 
1 .2 
1.8 
2.4 
3-2 
4.0 

5-o 

7.2 

9.9 
12 .9 


•39 

.69 

1 .1 

i-7 

2 .1 

2.9 

3-5 

4-3 
6.2 

8-5 
11 .1 


•36 
64 

1 .0 

14 , 
1.9 

2-5 

3-2 

4.0 

5-7 

7-8 

10.2 


I. 
I 

2 

3 
3 
5 
7 
9 


33 

59 

92 

3 

7 

3 



7 
3 
2 

5 


•3° 
•53 
.84 

1 .2 

1.6 

2.1 

2.7 

3-3 
4-8 
6.6 
8.6 



WIRE-ROPE TRANSMISSION 

146. General. Ropes made of iron or steel wire have been 
used to a considerable extent for transmitting power over com- 
paratively great distances. The introduction of electrical trans- 
mission has, however, greatly curtailed the field as far as power 
transmission is concerned; although wire ropes are still much 



* See "The Blue Book of Rope Transmission," by American Mfg. Co. 



334 MACHINE DESIGN 

used for conveying materials such as coal, rock, etc., by means 
of buckets attached at intervals along the rope. The rope in 
such installations moves at very low velocities and constitutes a 
different problem from that of power transmission. Wire ropes 
are also much used for hoisting work such as elevator and mine 
work and for carrying static loads as in supporting smokestacks, 
masts and suspension bridges. 

147. Materials for Wire Ropes. Wire ropes are usually made 
of wrought iron, open hearth steel, or crucible steel. For very 
severe work especially strong crucible steel known as plough steel 
is used. For a few special cases, copper and bronze are employed. 

The John A. Roebling's Sons Co. publications give the fol- 
lowing values for the tensile strength of various kinds of wire. 

Swedish Iron 45,ooo to 100,000 lbs. per sq. in. 

Open Hearth Steel 50,000 to 130,000 lbs. per sq. in. 

Crucible Steel 130,000 to 190,000 lbs. per sq. in. 

Plough Steel 190,000 to 350,000 lbs. per sq. in. 

They also state that it is difficult to obtain from a sample of rope in 
a testing machine, more than 90 per cent of the aggregate strength 
of all the wires. This is due to the difficulty of getting a perfect 
grip on the rope so that all the wires will carry their full share of 
the load; and also because the inner wires of a strand are shorter 
than the outer wires and are therefore more quickly overloaded. 
The wires, on account of the twisted construction, also tend to 
mutually cut into each other, thus rendering them more liable 
to fracture under heavy loads. On account of this latter action 
ropes made with a short twist break at a lower percentage of 
their full strength than those of a longer twist. 

148. Power Transmission by Wire Rope. Wire ropes for 
power transmission are usually made of iron or soft steel and are 
laid up with a soft core of hemp in order to give greater flexibility. 
They cannot be run on metallic surfaces and the sheaves must be 
lined at the bottom with soft rubber or similar yielding material. 
Great care must be taken that the rope does not chafe and, un- 
like the sheaves for fibrous ropes, the grooves in sheaves used 
for wire rope are so formed that the sides of the groove do not 
compress the ropes. In wire-rope sheaves, the radius at the 



BELT, ROPE, AND CHAIN TRANSMISSION 335 

bottom of the groove is always greater than that of the rope 
itself so that wire-ropes drive, like flat belts, simply through 
the friction on the bottom of the groove, due to the tension of 
the rope. The lining of the bottom of the groove (leather, wood, 
or some other comparatively soft material) gives increased friction 
as well as less wear of the rope. The sheaves should be as large 
as possible to minimize the bending effect on the rope; one 
hundred rope diameters being often taken as the minimum 
diameter of the sheave. 

The general theory and equations developed for fibrous rope 
hold also for wire rope, proper constants being substituted. It 
is evident from this discussion that wire ropes can safely trans- 
mit a greater amount of power than fibrous ropes of the same 
diameter, because of the much higher allowable tensile stress. 

The table on the following page, which is taken from a circular 
of the John A. Roebling's Sons Co., shows the power that may 
be transmitted by iron ropes of various sizes with sheaves of 
different diameters and rotative speeds. These values are for a 
rope made with six strands around a hemp core, each strand 
consisting of seven wires. This table does not make allowances 
for the change of stress due to the change of centrifugal force at 
various speeds; but it does consider the influence of the sheave 
diameter on the bending stress. For example : a y%" rope on an 
eight-foot sheave running 100 r.p.m., transmits only 32 H.P.; 
while the same rope transmits 64 H.P., when running on a ten- 
foot sheave at 80 r.p.m. or at the same linear velocity. By re- 
ferring to Fig. 120 it is seen that a manila rope of 1%" 
diameter transmits only 30 H.P., at the most economical velocity, 
or at about twice the velocity in the above instance. 

For hoisting and for transmission, if the sheave diameters 
must be much smaller than those given in the preceding table, a 
more flexible rope is used. This consists of six strands around a 
hemp core, but each strand is made up of 19 wires, which are, of 
course, of smaller diameter than those used for corresponding sizes 
of seven-wire strands. The lining of the bottoms of the grooves 
in the sheaves should be maintained in good repair. If it be- 
comes irregular, through wear, the rope may be bent at a sharp 



33& 



MACHINE DESIGN 



TABLE XXII. 

TABLE OF TRANSMISSION OF POWER BY WIRE ROPES * 



+j 


t/5 


u 






+■> 


09 


u 






u 

ofo 


c 
o 


CD 

£ . 

£ a 
<- 




u 
0) 

is 



(1< 


0) 


c 


&8 


s . 

<< 






4) — 




opt 




0) 

to 

Ih 

O 

w 


9>— H 


£ > 
8 


<uPtf 
•a 

2° 

H 


.9*8 

Q 




CO 

O 


3 


8o 


2 3' 


3 

8 


3 


7 


I4O 


20 


9 

16 


35 


3 


IOO 


23 


3 

8 


,1 
3? 


8 


80 


J 9 


f 


26 


3 


120 


2 3 


3 

8 


4 


8 


IOO 


19 


f 


32 


3 


I40 


2 3 


3 

8 


4l 


8 


I20 


19 


f 


39 


4 


80 


2 3 


3 

5 


4 


8 


I4O 


19 


1 


45 


4 


IOO 


2 3 


3 

8 


5 


9 


80 


t 20 
' 19 


| A f 


5 47 
i 48 


4 


120 


2 3 


3 

5 


6 


9 


IOO 


j 20 
( J 9 


( _9_ 5. 
(16 8 


J 58 
t 60 


4 


140 


2 3 


1 


7 


9 


120 


( 20 
I 19 


I JL 1 

f 16 8 


] 69 
( 73 


5 


80 


22 


7 
1 (5 


9 


9 


140 


j 20 
( x 9 


| A 1 


( 82 
) 84 


5 


IOO 


22 


16 


11 


10 


80 


I18 


I ft tt 

I f 16 


S 64 
i 68 


5 


120 


22 


7 
1 6 


13 


10 


IOO 


ji9 
(18 


| t tt 


j 80 
I 85 


5 


140 


22 


7 
16 


15 


10 


120 


5 19 
1 18 


} 1 tt 


\ 96 
J 102 


6 


80 


21 


ft 


14 


10 


140 


{19 

1 18 


| f H 


i 112 
I 119 


6 


IOO 


21 


i 


17 


12 


80 


(18 

I 17 


I 11 1 

f 16 ,4 


j 93 
i 99 


6 


120 


21 


i 


20 


12 


IOO 


18 
I 17 


(.11 3 
f 16 4 


I 116 
\ 124 


6 


140 


21 


\ 


23 


12 


120 


18 
1 F7 


I 11 a 

f 16 4 


j 140 
( i49 


7 


80 


20 


9 

16 


20 


12 


120 


16 


I 


J 73 


7 


IOO 


20 


9 
T6 


25 


14 


80 


{? 


) 1 


J J 4i 
1 148 


7 


120 


20 


9 
T6 


30 


14 


IOO 


!■; 


} I I* 


j x 76 
1 185 



* Taken from a publication of the John A. Roebling's Sons Company, of 
Trenton, N. J. 

The above table gives the power transmitted by Patent Rubber-lined Wheels 
and Wire Belt Ropes, at various speeds. 

Horse-powers given in this table are calculated with a liberal margin for any 
temporary increase of work. 



BELT, ROPE, AND CHAIN TRANSMISSION 337 

angle in passing over the high spots of the lining, with a resultant 
increase in the stress of the wires. This last action, however, is 
not equivalent, so far as the life of the rope is concerned, to run- 
ning over a correspondingly smaller sheave, for every portion of 
each wire is bent around each sheave once during every circuit 
of the rope; while it is not likely that the same portion of the 
rope will frequently come in contact with any single irregularity 
in the lining. 

ROPES AND CABLES FOR HOISTING 

149. Fibrous Ropes for Hoisting. In power transmission it is 
usually possible to install sheaves large enough to prevent the 
bending action from seriously affecting the life of the rope; but 
in hoisting work this is not always possible, on account of the 
size and clumsiness of the resulting tackle. Thus, a manila 
rope of 1 inch diameter, if used for power transmission, should 
run over a sheave at least 40 inches in diameter but if used for 
hoisting it might be required to run over a block sheave 1 2 inches 
or even 8 inches in diameter. The internal friction and external 
chafing are, in such cases, very great and the life of the rope, 
even when working at a lower stress, is greatly shortened; but 
in hoisting tackle, the frequency with which any portion of the 
rope passes over the sheaves is much less than is ordinarily the 
case in power transmission, on account of lower speed. 

Theoretical considerations are of little or no help in hoisting 
installations, and recourse must be had to successful practice on 
which, fortunately, there are considerable data. The following 
table, from a paper presented by Mr. C. W. Hunt, before the 
A. S. M. E., gives the results of a long series of observations, and 
indicates the most economical size of rope for a given load. It 
has been found, by experience, that ropes larger or smaller than 
those recommended in the table are shorter-lived under the load 
indicated. The speeds indicated in the table are defined as 
follows : 

"Slow" — Derrick, crane, and quarry work; 50 to 100 feet 
per minute. 
22 



33« 



MACHINE DESIGN 



"Medium" — Wharf and cargo work; 150 to 300 feet per 
minute. 

"Rapid" — 400 to 800 feet per minute. 



TABLE XXIII 

WORKING LOAD FOR MANILA ROPE 



A. 


B. 


C. 


D. 


E. 


F. 


G. 


H. 










Minimum Diameter of Sheaves 


Diame- 


Ulti- 


Working Load in 


Pounds. 




in Inches. 




ter of 


mate 

Strength, 












Rope, 














Inches. 


Pounds. 


Rapid. 


Medium. 


Slow. 


Rapid. 


Medium. 


Slow. 


I 


7,lOO 


200 


400 


1,000 


40 


12 


8 


I* 


9,000 


250 


500 


1,250 


45 


13 


9 


ii 


11,000 


300 


600 


1,500 


50 


14 


10 


if 


I3.400 


380 


750 


1,900 


55 


15 


11 


i4 


15,800 


45° 


900 


2,200 


60 


16 


12 


if 


18,800 


53° 


l,IOO 


2,600 


65 


17 


13 


if 


21,800 


620 


1,250 


3,000 


70 


18 


14 



150. Wire Hoisting Ropes. On overhead travelling cranes, 
elevators and mine work, iron or steel cables are used almost 
exclusively, as here it is usually possible to install sheaves or 
drums of large diameter. For rough service, deep mine work or 
wherever great strength is necessary, these ropes are sometimes 
made of crucible or plough steel. Great care should be exercised 
in installing such ropes and it is well, in general, to obtain the 
advice of the manufacturers before selecting any rope made of 
crucible steel, especially if great safety is desired. A factor of 
safety of at least 5 should be used in ordinary work, and for 
elevator, or similar work, a factor as high as 10 or 15 is some- 
times desirable. Table XXIV. taken from a publication of the 
John A. Roebling's Sons Co., gives data on standard hoisting 
ropes. For open-hearth steel the strength as given for iron rope 
may be increased 25 per cent. It will be noticed that these 
tables are based on a factor of safety of 5. 

CHAINS AND CHAIN TRANSMISSION 

151. Chains may be conveniently divided into three classes: 

(a) Chains for raising and supporting loads. 

(b) Chains for conveying purposes. 

(c) Chains for power-transmission purposes. 



BELT, ROPE, AND CHAIN TRANSMISSION 



339 



Chains for Hoists. In the first class are such chains as are 
used on cranes and hoisting appliances. Chains of this character 
are made with elliptical-shaped links and should be manufactured 
of the best wrought iron to insure perfect welding where the link 
is joined. The links themselves should be as small as possible 
to minimize the collapsing action or bending due to the pull of 
the adjacent links, and also that due to winding the chain upon a 
circular drum. Such chains are sometimes called short-link, 
close, or crane chains. 



TABLE XXIV 

STRENGTH OF IRON AND STEEL HOISTING ROPES 



1 
Swedish Iron. 




Cr 


ucible Steel. 




. "^ 

o3 o 
g£ 

s 


ail 

<U o 


u 

g cBs 

gm£.S 
<: 


g w 


<- ° C 

.5 qj > a> 


.5 

»h C/3 

■^ si 
go 
g a 

Q 


<u 


g «•£ w 

gm£.g 


9fl 

•JH V) 

Jh c8 O 


g °.S 

3 & CO .! 

g£ So? 

■S § 8^ 


2 a 


n-95 


114 


22.8 


16 


2f 


n-95 


228 


45 - 6 


10 


2 2 


9-»S 


95 


18.9 


15 


24 


9 85 


190 


37-9 


94 


2 4 


8.oo 


78 


I5.60 


13 


2i 


8.00 


156 


31.2 


84 


2 


6.30 


62 


12 .40 


12 


2 


6.30 


124 


24.8 


8 


t 3 - 


4-85 


48 


9 .60 


10 


If 


4-85 


96 


19 .2 


71 


If 


415 


42 


8.40 


84 


I* 


4-i5 


84 


16.8 


6J 


i£ 


3-55 


36 


7 .20 


74 


i4 


3-55 


72 


14.4 


5i 


i* 


3.00 


3 1 


6.20 


7 


if 


3.00 


62 


12 .4 


54 


ii 


2-45 


25 


5.00 


64 


ii 


2-45 


5° 


10. 


5 


i£ 


2 .00 


21 


4.20 


6 


1* 


2 .00 


42 


8.4 


44 


i 


1.58 


17 


3 -4o 


si 


1 


1.58 


34 


6.8 


4 


7 
8 


1 .20 


13 


2 .6a 


4i 


7 

8 


1 .20 


26 


5-2 


34 


3_ 
4 


0.89 


9-7 


1.94 


4 


3 
4 


0.89 


19.4 


3.88 


3 


5 

8 


0.62 


6 


8 


1.36 


34 


1 


0.62 


13 


6 


2 .72 


2i 


9 
16 


0.50 


5 


5 


1 .10 


a* 


9 

16 


0.50 


11 





2 .20 


if 


4 


°-39 


4 


4 


0.88 


a* 


4 


°-39 


8 


8 


1 .76 


i4 


tV 


0.30 


3 


4 


0.68 


2 


7_ 
16 


0.30 


6 


8 


1.36 


i* 


1 


0.22 


2 


5 


0.50 


i* 


3 

8 


.22 


5 





1 .00 


1 


5_ 
16 


015 


1 


7 


o-34 


1 


5 

16 


0.15 


3 


4 


0.68 


2 


i 


O.IO 


1 


2 


0.24 


3. 

4 


1 
4 


.10 


2 


4 


0.48 


4 



The strength of a chain link in tension is less than twice the 
strength of a bar of the iron from which the chain is made on 
account of the bending action due to the manner in which the 
load is applied, and also on account of the weld. If W — the 
breaking load in pounds, and d — the diameter in inches of the 



340 MACHINE DESIGN 

bar from which the link is made, then the following empirical 
equation may be used for iron crane chains. 

W = 54,000 d 2 (1) 

The working load (W) should not be more than one-third this 
value or W = 18,000 d 2 (2) 

In many cases a lower stress than indicated by (2) should be 
adopted. Whenever the load is not a direct pull, but severe 
bending stresses are also induced, as in chain " slings" for handling 
heavy iron castings, the chain should have great excess of strength. 
Chains should be carefully inspected and tested or " proved" 
before using. The " proof " usually applied is one-half the ulti- 
mate load. Where chains are used for hoisting work, they are 
likely to become badly strained. Annealing by heating allows 
a readjustment of the structure of the iron, and this should be 
done periodically with all such chains, particularly chains used 
for slings. This also affords an opportunity to thoroughly inspect 
chains which are greased in operation. The uncertainty regard- 
ing the exact condition of a chain in service, and the fact that it 
gives no warning of weakness, but may break at a load below the 
normal working load, have caused them to be largely replaced, on 
such appliances as overhead cranes, by steel rope. The state 
of the strength of the latter is more easily determined by inspec- 
tion. 

Weldless* steel chain rolled from a bar of special shape 
has lately come into use to some extent. The chain is made in 
lengths of from 60 to 90 feet, and the lengths are joined together 
by a link made of special welding steel. They are said to be 
much stronger than iron chains. 

152. Chain Drums and Sheaves. Drums on which crane 
chains are to wind should be carefully grooved so that alternate 
links lie flat on the surface of the drum; and should have sufficient 
capacity to receive the chain in one layer, as overwinding brings 
severe stresses on the parts wound upon the drum. The diam- 
eter of the drum should in no case be less than twenty times the 
diameter of the chain used, and thirty times this diameter is better. 

* See "Machine Design," by H. J. Spooner, page 452. 



BELT, ROPE, AND CHAIN TRANSMISSION 



341 



Where it is not possible to have the chain wind upon a drum, 
pocket chain wheels are often used. These wheels are made with 
pockets around the periphery into which the links fit. The 
links are prevented from coming out by a guide over a portion of 
the wheel; and hence cannot slip on the sheave. Anchor chains, 
and the chains of certain forms of chain blocks for raising weights, 
run over such sheaves. 

153. Hoisting-Hooks. The hooks used for raising heavy 
weights deserve special attention. They are usually made of 




Fig. 123 (b). 



steel or iron forgings although steel castings are employed to 

some extent. If the stress in the hook can be kept low the use of 

steel castings may be justified; but where the load is great and 

the fibre stress in the hook necessarily high, to avoid clumsy 

proportions, the hook should be forged from ductile material. 

Let the hook in Fig. 123 (a) be subjected to a vertical load P; 

then XY, the most dangerous section, is apparently acted upon 

p 
by a direct stress p' = — (where A is the area of the section) and 

by a rlexural stress p" due to the moment Pa; the stress p" being 



342 MACHINE DESIGN 

tensile at X and compressive at F. The theory of article 19, 
therefore applies, apparently, and equation M (Table VI) may 
be used to design the section, or 

P Pae 
p= p> + f = _ + _ _ 

Experience shows that members of this kind, even when 
made of materials whose tensile and compressive elastic limits 
are about the same, almost invariably yield to rupture on the 
tension side; and the section is usually made as shown, the 
gravity axis being located nearer the load, thus decreasing the 
tensile stress and increasing the compressive stress, as computed 
by the above equation. Recent investigations* have shown that 
in a curved beam loaded in this manner, the neutral axis does 
not coincide with the gravity axis, as in straight beams, but is 
located nearer the tension side, and the above theory is therefore 
defective, as the true tensile stress is greater than that given by 
equation M. That this is true is borne out by the fact men- 
tioned above, that hooks fail in tension when designed with an 
apparent compressive stress considerably above the tensile stress, 
although the elastic limit for either stress is about the same. 
The application! of the more accurate theory is, however, some- 
what complicated and it is believed that equation M may be 
safely used if due care is taken in assigning the limits of stress. 

Hooks for small cranes and hoists are much more likely to be 
loaded frequently to their full capacity than hooks for raising 
large loads; thus a hook on a five-ton crane may be loaded to its 
full capacity several times every day, while the hook of a twenty-ton 
crane would be thus loaded at rare intervals. The stresses in 
small hooks must therefore be kept low, and fortunately this can 
be done without making the hook clumsy. As the size of the 
hook increases, however, the stresses must necessarily be in- 
creased to avoid clumsiness, but the larger the hook the less 
frequently will it be fully loaded and a working stress as high as 

* See "Strength of Materials," by Slocum and Hancock. 

f See " On a Theory of the Stresses in Crane and Coupling Hooks. With Ex- 
perimental Comparisons with Existing Theory," by Professor Karl Pearson and 
Mr. E. S. Andrews. Messrs. Dulan & Co. 



BELT, ROPE, AND CHAIN TRANSMISSION 



343 



15,000 pounds per square inch, or more, is as safe in a fifty- ton 
hook as 10,000 pounds per square inch would be in a ten-ton 
hook. (See Art. 25.) 

The most valuable data on crane hooks is that given by Mr. 
Henry R. Towne in his " Treatise on Cranes," as a result of both 
mathematical and experimental work. Fig. 123 (a) and the 
following formulae give the most important dimensions of a hook 
according to this work, and these proportions have been much 
used with uniform success. The basis for each size is a com- 
mercial size of round iron or dimension A. In the following 
formula A is the nominal capacity of the hook in tons of 2,000 
pounds. The dimension D is assumed arbitrarily but so as 
to provide ample room for the slings. The following measure- 
ments are then expressed in inches: 

D = .5 A + 1.25 H = 1.08 4 K = 1.13A 

G = .jsD 7 = 1.33,4 £ = 1.05,4 

The following gives the capacity of the hooks made from 
various sizes of bar stock : 



TABLE 


XXV 


















Capacity of hook in tons 


1 


1 
f 


1 
2 


I 


i* 


2 


3 


4 


5 


6 


8 


10 


Size of bar A in inches 


5 

8 


11 
16 


3 

4 


11V 


ii 


if 


T 3 


2 


2\ 


z\ 


■i 


3i 



It is to be noticed that the stresses allowed by Mr. Towne's 
proportions are very low. Thus in a ten-ton hook the dimension 
A is 3X" or, allowing for finishing, the dimension B may be taken 
as 3", which would give a tensile stress in the shank of only 3,000 
pounds per square inch. It should be borne in mind, however, 
that hooks are subjected to much abuse and the designer has no 
assurance that they will always be loaded with a true axial load, 
for improper arrangement of the sling often throws the load more 
toward the point of the hook and the member is called upon to 
carry a bending moment greatly in excess of that for which it is 
intended. 

When, however, hooks larger th*.n those covered by Mr. 
Towne's work are to be designed his proportions lead to clumsy 
dimensions. Thus a twenty-ton hook would require a shank 



344 



MACHINE DESIGN 



4X" in diameter and a fifty-ton shank would be 6%" in diameter. 
Fig. 123 (b) shows a twenty-ton hook of Norway iron which has 
been successfully used in practice. The threaded shank being 
^yi" in diameter is therefore stressed to about 6,000 pounds per 
square inch but yet is only as large as the shank of a ten-ton hook 
as given by Mr. Towne's dimensions. Examination of current 
practice and measurements taken from a number of large hooks 
in successful service indicate an allowable tensile stress at X, 
as computed by equation M of Table VI, ranging from 10,000 
lbs. per square inch in ten-ton hooks, to 15,000 lbs. per square 
inch in fifty-ton hooks. 

154. Conveyor Chains. Chains for conveying and elevating 
materials, such as grain, coal, ashes, etc., are usually made of 




Fig. 124 (a). 



Fig. 124 (b). 



Fig. 124 (c). 



malleable iron, the links hooking together in some manner. This 
style of chain is known as link belt. On account of the diverse 
purposes to which they are applied, they are made in many forms, 
and the selection of the particular form for a given problem is 
usually made in conference with the manufacturer or taken from 
trade catalogues giving the desired information. This form of 
chain is also extensively used in rough machinery, such as agri- 
cultural implements, for the transmission of power. In such 
cases the chains must be run at low speeds, as they become noisy 
and unreliable even at moderate velocities. 

155. Chains for Power Transmission. The chains heretofore 
discussed move, necessarily, at low velocities, but of late a demand 
has arisen for chains which may be run at high speeds for the 
purpose of transmitting power. Such chains are used when a 



BELT, 



ROPE, AND CHAIN TRANSMISSION 



345 



positive velocity ratio must be maintained between the connected 
shafts, and where the distance between shafts is so great as to 
make tooth gearing inconvenient. Of this class there are at 
present three principal types, namely, roller chains, block chains, 
and so-called silent chains. Fig. 124 (a) illustrates the simplest 
form of roller chain in which the pin A is riveted fast in the outer 
links, and rotates in the inner links. The roller R lessens the 
friction against the tooth. In this form of chain the wear between 
the pin and the inner link is excessive, and for this reason it is 
now little used for power transmission. It is sometimes made 
without the roller and with several inside links and is then known 
as stud chain. In this form it is used for very low velocities only. 




Fig. 125. 



Fig. 126. 



The form shown in Fig. 124 (b) is most common. Here the bush- 
ing B is pressed into the inner links, and the pin, which is riveted 
fast to the outer links, bears over the whole length of the bushing. 
The roller R rotates on the bushing. In the block chain, Fig. 
124 (c), the pin also bears over the whole thickness of the block 
D, but since the roller is necessarily omitted, there is more friction 
against the tooth. Roller chains may be used for velocities up 
to about 800 feet per minute, and block chains up to about 500 
feet per minute. 

The defect in the operation of the roller or block chain may 
be seen by referring to Figs. 125 and 126. When the chain is 
new, and has the same pitch as the wheel, it fits down on the wheel 
as shown in Fig. 125, but in a very short time the chain stretches 



346 



MACHINE DESIGN 



slightly, due to wear of the joints, thus increasing the pitch of 
the links. The wheel, on the other hand, may wear but this does 
not change the pitch. The operation of the chain is then as shown 
in Fig. 126, the increased pitch causing the rollers to ride higher 
and higher on the back of the tooth as they move round the 
sprocket. The roller A is shown fully seated while B is just 
coming down to its seat. Before B can become fully seated A 
must rise, and this action takes place when A and B are carrying 
full load. As a consequence the chain does not run quietly and 
smoothly and the wear is excessive, thus limiting the speed at 
which the chain may be run. This difficulty is sometimes met 
by the arrangement shown in Fig. 127. Here the pitch of the 



Tight Side 





Slack Side 



Fig. 127. 

chain when new is made a little less than the pitch of the driving 
sprocket, and clearance is allowed between the roller and the 
tooth, so that the driving is done by the last tooth L; the pitch 
of the chain being such that the incoming roller M just clears the 
back of the first tooth and seats itself close to it at the root as at 
N. As the chain stretches, the rollers move backward toward 
the faces of the teeth, till a condition like that in Fig. 125 is reached, 
and riding commences. The pitch of the driven sprocket wheel 
is made equal to that of the chain, and the condition when new 
is that shown in Fig. 127. As the chain stretches, the rollers 
move gradually backward away from the driving faces of the 
tooth, the driving being done on the last tooth P. It is evident 



BELT, ROPE, AND CHAIN TRANSMISSION 347 

that this construction extends the time preceding the condition 
shown in Fig. 126. When this construction is used, the form of 
the tooth must be slightly modified. Referring to Fig. 128 it is 
obvious that if the outline of the tooth M be an arc of a circle 
struck from the centre of the roller (2), this roller will swing from 
its position i' rolling on the face of the tooth, and this is the 
usual outline. But before roller (3) can take the load, which (2) 
is about to give up, it must be fully rooted against the next tooth ; 
whereas (from Fig. 128), a small distance now separates the two. 
Therefore as (2) rolls up the curve of the tooth it should allow 
(3) to slowly settle back in place. The tooth outline is therefore 
struck (as shown on M), from a point a little inside the pitch 
polygon so as to give a curve tangent to the first and last positions 




Fig. 128. 

of the roller. This outline is also necessary for the back of the 
tooth in order to allow the incoming roller to swing in without 
striking. The velocity of the chain is, therefore, a little less than 
the theoretical velocity on account of this continual slipping back- 
ward. Brief reflection will show that the tooth outlines of the 
driven sprocket may be struck from the centre of the roller when 
rooted in place; and that when the chain is stretched a little it 
will creep as it is wound upon the driven sprocket. 

When the roller (4), Fig. 127, is about to roll up the face of 
P, roller (5) is not in contact with M (wear having begun); 
hence the chain will move ahead till (5) is in full contact with M. 

The greatest defect in this construction is the fact that the 
load is carried entirely on one tooth and hence the wear is ex- 



348 



MACHINE DESIGN 



cessive. This may be so great that the chain creeps forward on 
the driven wheel so as to cause the incoming roller to strike the 
tooth S, Fig. 127. 

The above difficulties are overcome in the so-called silent 
chains. In these chains the inevitable stretching of the links 
is compensated for in a peculiar manner. The true theory of the 
action of these chains is very complex; but the general action is 
as follows: — as the chain stretches, the links continually tend 
to take up a position farther and farther away from the centre of 
the sprocket; thus increasing the length of the sides of the pitch 
polygon to suit the elongation of the link. Each link therefore, 
remains in constant contact with its own tooth, from the time of 




Fig. 129. 

engagement till release takes place. The links seat themselves 
without sliding action and the operation is nearly noiseless. 

In the Renold chain of this type, Fig. 129, the links move 
relative to each other on a round pin P, the shouldered ends of 
which are riveted into a washer W, thus holding the chain together. 
In a later form half bushings of bronze are so fitted to the links 
that the pin has a bearing over its full length; but the relative 
motion of the pin to the bush is still a sliding motion. In the 
Morse chain this sliding is eliminated by an ingenious form of 
rocker joint shown in Fig. 130. The hardened steel parts, A 
and B, are fitted respectively to the sets of links, D and C. While 
keeping contact along a fixed line they rock on each other as the 
links C and D move relatively to each other, and sliding is thus 



BELT, ROPE, AND CHAIN TRANSMISSION 349 

eliminated. When transmitting simple tension between the 
sprockets, the parts A and B are in contact on flat surfaces as 
shown at E. This construction has the advantage of requiring 
little or no lubrication, hence the chain maybe run at higher speeds 
than others requiring lubrication, the speeds of which are limited 
by the velocity at which centrifugal action throws off the lubri- 
cant. The Morse chains also work well in dusty places. 

The efficiency of both of these chains is very high, the makers 
of the Morse chain claiming an efficiency of nearly 99 per cent. 




Fig. 130. 

Such chains are particularly useful for connecting shafts which are 
too far apart for gearing, and not far enough for a belt, and in 
places where positive connection is desirable, as in motors 
driving heavy machine tools. It is to be especially noted that 
this form of transmission requires no definite tension on the 
slack side of the chain to produce a certain driving force on 
the tight side; and hence the pressure* on the bearings is much 
reduced, for a given effective pull on the wheel rim. 



CHAPTER XIII 
APPLICATIONS OF FRICTION 

Friction Wheels for Power Transmission 

156. General Considerations. When it is required to drive 
a rotating member intermittently, and the rate of driving is 
not necessarily positive, friction wheels have been found very 
useful. They are particularly applicable where the amount of 
power is comparatively small, as in feed mechanisms, but they 
may also be used for heavy work when properly constructed. 
For continuous driving the transverse sections of friction wheels 




-£ 



B 






\ 




=A= 



m^Sl^ 



§ 



Fig. 131. 



Fig. 132. 



Fig. 133. 



must be circular in cross-section, and this form, only, is used in 
practice. 

Figures 131 and 132 show common forms of friction wheels. 
In Fig. 131 let A be the driving wheel which rotates continuously, 
and let B be the driven wheel which is required to be driven 
intermittently. The shaft of A is so mounted that, by means of a 
lever attached to the bearing, A may be pressed up against B 
with a force P, or it can be moved slightly away from B until no 
contact exists. If now the force P is applied to the bearing 
(which should be close to A), an equal and opposite force is set 
up in the bearing of B, and the wheels are pressed together at the 

35° 



APPLICATIONS OF FRICTION 



351 



line of contact. The resistance to slipping at the line of contact 
will be fiP, where /i is the coefficient of friction of the materials 
of which the wheels are made; and if julP is equal to, or greater 
than, the resisting force at the surface of B, A will cause B to 
rotate. Theoretically, A and B will roll together with pure rolling 
motion, but practically this cannot be attained, as even with very 
hard materials the wheels flatten slightly at the line of contact. 
(See Art. 108.) 

Fig. 132 illustrates the application of friction wheels to shafts 
which are not parallel to each other, the wheels here having the 
form of rolling cones. Obviously the principle is of wide applica- 
tion and many combinations of friction wheels are used. Fig. 
133 illustrates a friction wheel arranged so that the driver A 




Fig. 134. 



Fig. 135 (b). 



Fig. 135 (a). 



can rotate the driven wheel B in either direction, depending on 
whether it is pressed against the surface m or the surface n. 

Fig. 134 shows a form of friction mechanism much used for 
imparting variable speed to the driven shaft. The driver A 
may be moved along the shaft C at will. When at A' the angular 
velocity of B is a minimum. As A is moved inward, the rotative 
velocity of B increases. When A is moved across the centre of B 
to the other side, the direction of rotation of B is reversed. If 
A were infinitely thin, it would, theoretically, roll upon B with 
pure rolling motion. Since, however, it must have an appreciable 
width of face, and since the velocity of B varies with the radius, 
it is evident that there must be some sliding at the line of contact. 
For this reason the thickness of A must, for best results, be kept 
small compared to the radius of B. 



352 MACHINE DESIGN 

157. Materials for Friction Wheels. The driven wheels of 
friction devices should always be made of a harder material than 
the driver, for the reason that the driven wheel is likely at any 
time to be held stationary by the load, while the driving wheel 
revolves against it under pressure. This action, while severe 
on the driver, does not tend to wear it out locally, while it does 
rapidly wear flat spots on the driven wheel. Driven wheels are, 
therefore, almost universally made of iron, and driving wheels 
of wood, leather, paper, rubber or of some composition of 
these; the most common being leather and various forms of 
paper. 

158. Practical Coefficients. The tangential force F, exerted 
by A upon B, Fig. 131, is dependent on the pressure P and the 
coefficient of friction //. It is, therefore, necessary to know the 
allowable pressure per unit of length along the contact elements 
and also the value of /* for the particular materials used. The 
most comprehensive investigation of these relations is that made 
by Professor Goss,* whose experiments cover a variety of materials. 
He recommends the following pressures, which are about one-fifth 
of the ultimate crushing strength of the respective materials. 

Safe Working Pressures per Inch of Contact. 

Material. Pressure. 

Straw fibre 150 

Leather fibre 240 

Tarred fibre 240 

Leather 150 

Woodf 100 to 150 

Professor Goss found that the coefficient of friction for all the 
wheels tested approached a maximum value when the slip be- 
tween the two wheels was about 2 per cent, and, within narrow 
limits, was practically independent of the pressure of contact. He 
found these values to range for different combinations from low 
values up to .515. In these experiments the friction due to the 
bearings was neglected. The bearings, however, were of the 
roller type and, probably, absorbed less power than the ordinary 

* See Transactions A. S. M. E. Vol. XXIX. 

t The value for wood is not from Professor Goss's paper. 



APPLICATIONS OF FRICTION 353 

bearing. Making due allowance for the difference between 
laboratory conditions and those found in practice, Professor 
Goss recommends the following approximate values of // * for 
the various combinations. In this connection it is to be noted 
that allowance must be made for a decrease in the value of this 
coefficient when the linear velocity of the driver is great, in 
the case where the driver is starting the driven wheel under 
load (see Art. 28). 

Working Values of Coefficient of Friction. 
Materials. Coefficient of Friction. 

Straw fibre and cast iron 0.26 

Straw fibre and aluminum 0.27 

Leather fibre and cast iron 0.31 

Leather fibre and aluminum o . 30 

Tarred fibre and cast iron o . 15 

Tarred fibre and aluminum o . 18 

Leather and cast iron 0.14 

Leather and aluminum 0.22 

Leather and typemetal 025 

Wood and metal 0.25 

159. Power Transmitted by Friction Wheels. If V be the 

velocity of the surface of the friction wheels in feet per minute, 

P the total normal pressure in pounds, F the resulting tangential 

force, and p. the coefficient of friction; then since F = jj.P, the 

rate at which power is transmitted in foot pounds per minute is 

/lP V, and the horse-power is 

aPV 
HP = ^ (1) 

33,000 

of if d be the diameter of the driver in inches, / the length of face 
in inches, w the allowable load per inch of face, and N the number 
of revolutions per minute, the horse-power is 

fj-wl X 7T d N 

H. P. = = 0.000008 awl d N . . (2) 

12 X 33,000 

Example. How many horse-power can be transmitted by a 
straw-fibre friction pulley of 8" diameter and 6" face, when run- 
ning at 500 r.p.m., the driven wheel to be of cast iron? 

* The coefficient for wood is not from Professor Goss' paper. 
23 



354 MACHINE DESIGN 

Here d = 8", I = 6", AT" = 500, \i = 0.26, w = 150 

. * . H. P. = .000008 X 0.26 X 150 X 6 X 8 X 500 = 7.5 

It may be noted that the horse-power per inch of width of 
face is a little more than unity, for a surface speed of 1,000 feet, 
as in the above example. This corresponds closely to the empiri- 
cal rule given for belts in Art. 136; and corroborates the empirical 
rule often used that the same width of face is necessary for a 
friction wheel as for a belt, to transmit a given horse -power at 
the given speed. 

In the case of bevel wheels (see Fig. 132) the component R 

P 

of the applied force P presses the wheels together and R = 

cos 6 

The velocity of the mean circumference of the driver may be 
taken as the velocity of transmission. 

In face friction driving as in Fig. 134, the width of the driving 
wheels should be kept as narrow as possible for best results. 
If the velocity of the outer edge of the driving wheel is not 
more than 4 per cent greater than that of the inner edge, the 
above coefficients may be used. Where the driver must, at times, 
drive at a short distance from the centre, lower values of the 
coefficient of friction must be taken. 

The faces of a pair of metal friction wheels are sometimes 
formed as shown in Fig. 135 (a), and are then known as wedge- 
faced friction wheels. The object of this construction is to 
secure a greater resistance to slipping, with a given radial pressure. 
It is to be noted that the number of wedges does not affect this 
ratio, but decreases the wear by distributing it over several sur- 
faces. This last item is important, as it is easily seen that the 
contact surfaces of the driver and the driven wheel can have the 
same velocity at one point only, and that at all other points 
slipping or a grinding action occurs and wear must result.* The 
teeth therefore should not be very long. 

In Fig. 135 (b), let P be the radial force applied to the wedged 

surface, F the tangential force transmitted, — the reaction on 

* See "Kinematics of Machinery," by John H. Barr, page 106. 



APPLICATIONS OF FRICTION 355 

each face and 26 the angle of the wedge; then the wedge is held 

in equilibrium by the force P, the reactions — and the frictional 

2 

resistances ft — due to the wedging action. Equating vertical 

forces 



P = 2 I — s in 6 + cos 6) or since F = 2 I ) 



or£ = - 

F sin 
P = + F cos . . . . (1) 

ft v ' 

HP 

or F = — (2) 

sin + u cos 6 



To avoid sticking the angle 2 should not be less than 30 . 



FRICTION BRAKES 

160. Friction brakes are used for controlling and stopping 
machinery by absorbing energy through frictional resistance from 
some moving part, and dissipating it as heat. Brakes used in 
heavy work, and as dynamometers for measuring energy, must 
often be fitted with water circulation to carry away the heat. 
The student is referred to treatises on power measurement for 
a discussion of dynamometers. 

161. Block Brakes. The simplest form of brake is the block 
brake as shown in Fig. 136. Here the force P, acting on the 
lever A, presses the block C against the wheel B. Let the reaction 
between the wheel and the block be R. Then if B be rotating, a 
tangential frictional resistance /tR = F will oppose its motion. 
With the arrangement shown in Fig. 136, the line of action of F 
passes through O the centre of the fulcrum of A. Considering A 
as a free body and taking moments around O, then for rotation 
in either direction 

F 

P (a + b) = Rb or since R = — 

Fb 
F ~V(a+b) (I) 



356 



MACHINE DESIGN 



In Fig. 137 the line of action of F does not pass through O and 
therefore in writing the equation for the equilibrium of A its 
effect must be considered, whence 

Fb 



P = 



n 



1 



(2) 



The minus sign is to be used for rotation in a clockwise 
direction, for the arrangement shown, and the plus sign for rota- 




Fig. 137. 



Fig. 138. 



tion in the opposite direction. It is to be especially noted that 

1 c 
for clockwise rotation when — = — , or when b = a c y P = o*: 

that is, the brake is self-acting and if put in contact the moment 
of the frictional force will apply it with ever-increasing pressure. 
Obviously such proportions should be avoided. 
In a similar manner for Fig. 138 

Fb r 1 c-i 

the plus sign referring to clockwise rotation, jor the arrangement 
shown, and the minus sign to rotation in the opposite direction. 
In this class of brakes the pressure of the brake R against 
the wheel is opposed by an equal force R f at the bearing near the 
wheel. In the calculations above, the braking effect due to 
friction of the journal is neglected, as its lever arm is, usually, 
small. It cannot be neglected in designing the bearing, and for 
this reason this form of brake is not well adapted to heavy work. 



APPLICATIONS OF FRICTION 



357 



Fig. 139 shows an arrangement of brake beams for heavy work 
such as is used in mining machinery. The force W, which may 
be applied by a steam cylinder, acting on the system of levers, 
causes the brake beams B&ndB to press equally on opposite sides 
of the wheel, and causes no pressure on the bearings of the drum. 

T 

If — be the tension in each of the rods A and ;!, the frictional force 
2 

exerted on the wheel is F = 2 i± T. If the pin O is so located 
that when the load W is applied it moves to O', and the centre 




Fig. 139. 



line of the rod A passes through the centre of the pin P, a toggle 
effect is obtained and the tension in the rods A and A may be made 
any desirable value; in fact with such an arrangement care must 
be exercised in adjusting the brake that such pressures are not 
brought on the pins as will cause failure by shearing. When O 
moves down to O f the brake is "locked" in position and the 
operating force may be removed. This last feature is often a 
valuable quality in a brake. Brakes of this character are gener- 
ally lined with wooden blocks as shown. 

162. Strap Brakes. If the effect of centrifugal force is neg- 
lected (see Art. 131), and the total tensions in the band (T 1 



358 



MACHINE DESIGN 



and T 2 ) be taken instead of the tensions per inch of width, equa- 
tion (6) of that article reduces to 



= 10 



(i) 



Where k = 0.0076 pt a, a being the arc of contact in degrees. If, 
also, F is the total frictional force exerted by the band upon the 
wheel, 



T — T 



(2) 



It is obvious that these equations are applicable to the discussion 
of band brakes. Figs. 140, 141, and 142 show the most usual 
arrangement of band brakes. In Fig. 140 the end of the strap 




Fig. 141 



Fig. 142. 



which is subjected to the greatest tension 7\ is anchored, for con- 
venience, at the pin which serves as a fulcrum for the operating 
lever L; it could be anchored to any other convenient part of the 
frame. 

F 
From (1) and (2), T 2 = 



10 



Taking moments around O 



Pa = T 2 b 



ot P = 



Fb 



10 
Fb 



a(io k -i) (3) 

which expresses the relation between the applied force P and the 
frictional resistance applied to the wheel. 



APPLICATIONS OF FRICTION 359 

In Fig. 141 the end under greatest tension is attached to the 
lever and the end of least tension is anchored, hence for this case 

n FJr io k 1 

In Fig. 142 the end under greatest tension is anchored to the lever 
at a shorter radius than the end of least tension; so that the force 
which it exerts assists the operating force P. This is known as 
a differential brake. For this case in a similar manner as above 






It is to be especially noted that if io k b 1 = b 2 , P = o, and 
the band will brake automatically; that is if any force is applied 
to the lever, the brake will continue to set itself up with ever 
increasing force till motion ceases or rupture occurs. This form 
of brake is exceedingly dangerous on account of its tendency to 
"grab," especially if jj. is materially increased through a change 
in the character of the friction surfaces. 

Strap brakes are usually made of wrought iron or steel. In 
light work they may engage with a cast-iron surface or may be 
lined with leather; but in very heavy work they should be lined 
with wood. 

FRICTION CLUTCHES AND FRICTION PULLEYS 

163. Friction clutches though made in a great variety of forms 
can, in a large measure, be classified under four principal types, 
namely, Conical, Radially Expanding, Disc, and Band. A well- 
designed clutch should start its load quickly but without shock, 
and should disengage quickly. It should be "self-sustained," 
that is, when the clutch is driving, no external force should be 
necessary to hold the contact surfaces together. In addition, it 
is often necessary that the clutch should "lock" in place, after 
the manner of the brake in Fig. 139. 

164. Conical Clutches. Fig. 143 shows the elements of a 
conical clutch which is self-sustained. The cone F is fast to the 
shaft 5 and rotates with it. The pulley H rotates upon F and 



36o- 



MACHINE DESIGN 



carries with it the levers E. When the thimble B is forced under 
the rollers C, the levers E force the cone surfaces in contact. 
Heavy springs at G (not shown) throw the surfaces apart when 
the thimble is withdrawn. The relation between the trans- 
mitted frictional force F and the force P applied to the cone, in a 
direction parallel to the axis, is the same as that of the wedge 
gearing in Art. 159, or 

pP 

F = — (6) 

sin + P- cos 6 

The angle should not be less than io°, unless some mechanism 
is provided for separating the cone surfaces, positively, when 




Fig. 143. 



Fig. 144. 



desired. For clutches that do not operate frequently, metal 
surfaces are often used; but where the operation of clutching is 
frequent, one surface is usually lined with wood, cork, or leather. 
165. Radially Expanding Clutches. Fig. 144 shows the ele- 
ments of a radially expanding, self-sustained clutch. The clutch 
body A is keyed to the shaft, while the pulley C rotates loosely 
upon the shaft. The circular segment -B, which fits the inner 
surface of C, can be moved radially upon A . The loose ring G 
is operated axially by a forked lever fitting on the pins P. When 
the sleeve E is forced inward by the ring G, the links D force the 
segments B outward against C. In the arrangement shown the 
links have a toggle effect and can exert enormous pressure against 



APPLICATIONS OF FRICTION 36 1 

B, hence adjustment must be carefully performed. This is 
usually accomplished by making the length of the link D adjust- 
able, by means of turn-buckles or similar devices, which also 
provide a means of compensating for wear. Usually the sleeve 
has motion enough to carry the inner end of the link slightly past 
the centre position shown, thus locking the clutch in place. 

166. Disc Clutches. Fig. 145 shows the elements of a multiple- 
disc clutch as sometimes used in automobile work for connecting 
the engine to the transmission shaft, A being fast to the engine 
shaft and B to the transmission shaft. The part A carries a 
number of discs, C, which fit loosely in a radial direction but are 
prevented from rotating relatively to A by bolts E which also 
hold L, the cover of the case, in place. A second set of discs D, 
placed alternately between the discs C are carried on the part B 
and compelled to rotate with it by the keys G. A heavy helical 
spring F (sometimes made of rectangular section as shown) 
presses the two sets of discs together with a known load P, when 
the clutch is in and the shafts connected. The sleeve B while 
compelled by the feather S to rotate with the transmission shaft 
A 7 , can be moved axially by means of the grooved collar / and the 
ring J; I being made fast to B but built separately from it for 
constructive purposes only. When B is moved to the right the 
spring is compressed and the pressure on the discs relieved. The 
discs often run in an oil bath to prevent " grabbing." It is 
readily seen that while the force, P, which presses each pair of 
contact surfaces together is the same, the total frictional force 
transmitted is proportional to the number of pairs of contact 
surfaces n or 

F = 11 nP (7) 

If the mean friction radius of the discs be r, the frictional mo- 
ment transmitted is Fr = jmPr. In Fig. 145, n = 7. The 
above form of clutch is known as the Weston clutch. Obviously 
any number of pairs of discs may be used. For large work the 
discs are sometimes made of iron and wood (or wood-faced). 
For small work, alternate discs of steel and brass are employed. 
Many pairs of contact surfaces are then used and the discs run in 



362 



MACHINE DESIGN 



oil to prevent " grabbing." The width of the wearing faces of 
the discs should be made small to prevent undue wear toward the 
outer edges of the discs D, as in a thrust block (Art. 105). It is 
better to use more discs of a smaller diameter than a few of great 
face. 

167. Band Clutches. Fig. 146 illustrates the elements of a 
band clutch. The clutch wheel A (which may be fast to one 
shaft) carries the wood-lined band C. When the thimble F 
(which slides on the shaft) is forced under the lever E, the iron 
band C is tightened and clutches the rim of the driven wheel B. 




Fig. 145. 



Fig. 146. 



Obviously the principles involved are identical with those of the 
strap brake, Fig. 140 of Art. 162. For light work the band may 
be lined with leather, but in heavy work, such as mine hoisting, 
blocks of bass wood, or other soft wood, are used. The wood 
lining is usually made fast to the strap, though occasionally on 
very large diameters they are attached to the wheel so that they 
may be turned true in place. These clutches are made self- 
locking by arranging for a toggle effect in some one of the operat- 
ing levers. 

Occasionally the band is made to expand inside of the rim of 
the wheel to be driven. It is to be noted that this case is not 
the same as the one just discussed, but is a special case of a 



APPLICATIONS OF FRICTION 363 

radially expanding clutch. The outward force exerted by the 
band may be computed by the theory of Art. 78, considering 
the band as a thin cylinder under compression, the compressive 
stress at any section being that due to the pressure applied by the 
operating lever. 

168. Magnetic Clutches. A number of clutches have recently 
appeared which are operated magnetically. These are most 
generally of the disc type. Evidently the general principles above, 
regarding transmissive power, apply also to these clutches. In 
magnetic brakes, the load is usually applied by a spring, or 
weight, and released by magnetic action, thus insuring safety 
against accident should the electric service fail. 

Practical Coefficients for Brakes and Clutches. The most 
usual combinations of friction surfaces for brakes and clutches 
are wood, leather, or cork with iron; and iron with iron. In the 
multiple-disc type, brass or bronze on iron or steel are sometimes 
used. Mr. C. W. Hunt gives the following values of ft as the 
result of considerable experience in designing clutches, namely: 
cork on iron, 0.35; leather on iron, 0.3; and for wood on iron 0.2. 
For iron on iron jj. may be taken as 0.25 to 0.3. It should be 
remembered that if the friction surfaces are to be engaged at 
high velocity, lower values must be assumed than for lower 
speeds (see Art. 28). 

The pressure per unit area of surface is also an important 
feature in the design of friction machinery, for if this is taken 
too high, excessive wear will result. Thus in disc clutches the 
pressure is usually taken at not more than 25 to 30 pounds per 
square inch and lower values are desirable. Wooden surfaces 
should not be loaded beyond 20 to 25 pounds per square inch. 
If the clutch or brake is to operate frequently, ample surface 
must be provided to properly radiate the heat generated. 

References: — 

Transactions A. S. M. E., Vol. XXX, 1908. 
Transactions Inst. Mechanical Engineers, July, 1903. 



CHAPTER XIV 



TOOTHED GEARING 



169. General Principles. When it is necessary that rotation 
of one shaft shall produce definite and positive rotation of another, 
it is evident that friction wheels, as discussed in the preceding 
chapter, will not suffice where any considerable amount of power 
is to be transmitted. In such cases the peripheral surfaces of 
the transmission wheels are provided with teeth, so that the 
motion shall be positive. It is evident that any pair of surfaces 
which will roll together with pure rolling motion, so as to give the 
required velocity ratio, may serve as a basis for the design of a 
pair of toothed gears; and works on mechanism treat fully of 
the methods of drawing the sections of such surfaces for various 
conditions and velocity ratios. Whether the elements of the 
surface thus outlined shall be parallel or otherwise will depend on 
the angle which the shafts make with each other, as in the case of 
friction wheels, and tooth gearing may be classified * according to 
the character of the pitch surfaces, and the relation of the axes, 
thus: 



Kind. 


Relation of Axes. 


Pitch Surfaces. 


Spur 

Bevel 

Screw 

Skew 

Twisted 

Face 


Parallel 

Intersecting 

Not in one plane 

Not in one plane 

Any 

Any 


Cylinders 
Cones 
Cylinders 
Hyperboloids 
Any of the above 
None, strictly 



The most important of these are spur, bevel, and a few special 
forms of twisted and screw gears. The motion transmitted by a 
pair of properly designed toothed gears is identical with that 
of the base curves or surfaces rolling together. If r 1 and r 2 be 



* See " Kinematics of Machinery," by Professor C. W. McCord. 

3 6 4 



TOOTHED GEARING 



365 



the instantaneous radii of such a pair of surfaces at the point of 
contact, and u> x and «> 2 be their instantaneous angular velocities, 

o) t Y* 1 1 1 • 

then — = — . In the most common case the angular velocity 

of both shafts is constant and hence r ± and r 2 are constant, and 
the rolling surfaces are circular in cross -section. Thus Fig. 147 
shows a portion of two gears whose rolling surfaces are a pair of 
circular cylinders, represented in cross-section by the circles C 
and D. If the teeth are properly proportioned the motion trans- 
mitted will be identical with that produced by the rolling of C on 
D. It can be shown that the condition which such tooth outlines 




Fig. 147. 



Fig. 148. 



must fulfil in order that the velocity ratio may be constant, is 
that the common normal to the tooth outlines at the point of contact 
must always pass through the point of tangency of the rolling 
circles. There are many curves which can be used for tooth 
outlines, and which would fulfil the condition, but in practice 
only two are commonly employed, namely, the involute and the 
cycloid. 

Fig. 147 illustrates a portion of two gears with involute teeth. 
The upper wheel, M, is the driver. Contact between two teetrf 
has just begun at a, and the common normal to the point of con- 
tact a O b passes through the pitch point O. As the wheels rotate 
the point of contact will move along the line a O b till contact 
ceases at b. Hence in the involute system the normal to the 



366 MACHINE DESIGN 

point of contact makes a fixed angle with the common tangent to 
the pitch circles. 

Fig. 148 shows a portion of two gears with cycloidal teeth. 
Contact is just beginning at a, and as the gears rotate the point of 
contact will move along the curved path aO b, contact ceasing at 
b. The normal to the first point of contact is drawn, and it is 
clear that the inclination of the normal to the common tangent 
of the pitch circles, is a maximum at this point, and continually 
varies in direction though always passing through the point O. 
It can be shown that in the involute system the angular velocity 
ratio will remain constant, within the limits of action, whether 
the pitch circles are tangent or not; but for the transmission of 
constant velocity ratio with cycloidal gearing the pitch circles 
must remain tangent. The involute gear, therefore, has a decided 
advantage for general use and it has practically superseded the 
cycloidal for most work. A fuller treatment of the theory of 
gear-tooth outlines, which is beyond the scope of this work, will 
be found in treatises on mechanism.* 

170. Interchangeable Systems of Gearing: Standard Forms. 
It is desirable in practical work that any gear of a given pitch 
shall run properly with any other gear of the same pitch. In 
order that this may be so, certain limitations must be placed 
upon the form and dimensions of the tooth. In the cycloidal 
system interchangeability may be accomplished, as far as the 
tooth outlines are concerned, by keeping the diameter of the 
describing circle constant for all gears of the series. 

Any involute tooth outline will run properly with any other 
similar outline; and any gear with involute teeth will run with 
any other gear having similar teeth, as far as the length of the 
involute outlines will permit, providing the thickness of teeth 
will allow them to mesh. In order to obtain involute outlines of 
sufficient length, and a series of gears with fixed nominal pitch 
circles, the angle 0, Fig. 147, made by the line of action with the 
common tangent to the pitch circles must have a proper value, 
and be constant for all gears of the series. In the systems in 

* See "Kinematics of Machinery," by J. H. Barr, page in; also, " Machine 
Design," part i, by F. R. Jones. 



TOOTHED GEARING 367 

most common use this angle is 14K , though there is a ten- 
dency in modern work toward a greater angle. 

It is found undesirable in practice to make gears with less than 
twelve teeth ; and in some cycloidal systems the radius of a 
twelve-tooth gear of the required pitch is taken as the diameter 
of the describing circle. For a twelve-tooth gear this will re- 
sult in radial lines for the tooth outlines below the pitch circle, 
i.e., the tooth will have radial flanks. In the practice of the 
Brown & Sharpe Mfg. Co., the diameter of the describing circle 
is the radius of the fifteen-tooth gear of the series. This gives 
spaces between the flanks of the teeth on the twelve-tooth, or 
smallest gear, so nearly parallel that they may be cut with a 
rotary cutter. 

It is evident from Figs. 147 and 148 that the tooth outlines of 
any system may be extended both above and below the pitch line 
till they meet. It is also clear that the longer the teeth the earlier 
will they engage with each other, the greater will be the arc of 
contact, and the greater will be the number of teeth continually in 
contact. The distribution of the load over a number of pairs 
of teeth is in itself conducive to smooth running; but on the other 
hand, extending the arc of contact away from the pitch point, 
increases the sliding between teeth, and also the velocity with 
which the teeth approach each other. The tooth also becomes 
weaker as it is lengthened, the thickness remaining the same, 
and for these reasons a practical limit is placed on the length of 
teeth. The length of tooth adopted in practice is, therefore, a 
compromise between conflicting conditions, which experience 
has shown will give good results. 

The distance along the pitch line from any point on a tooth to 
a corresponding point on the next tooth, is called the circular 
pitch ; and will be noted by s. The thickness of the tooth along 
the pitch line will be denoted by t, Fig. 151. In the case of cut 

gears, where no clearance is allowed between teeth, t = — . In 

2 

some forms of gears, such as shown in Fig. 150, where a metal 

pinion engages with a gear having wooden teeth, the pitch may 

not be equally divided, but the metal tooth may be thinner than 



368 MACHINE DESIGN 

the wooden tooth. If N be the number of teeth and D the 

diameter, then evidently Ns = nD. If the number of teeth N 

be divided by the diameter, the quotient, or the teeth per inch 

of diameter, is called the diametral pitch and will be denoted by 

N . 71D n 

S. Since S = — and s = — -, S X s = x . \ S = - and 
D N s 

s = — . The diametral pitch is, ordinarily, the most convenient 

for use, and in this country practically all interchangeable sys- 
tems are based upon the diametral pitch. Thus a gear 24" in 
diameter and 3 diametral pitch would have 24 X 3 = 72 teeth, 

and the circular pitch would be — = 1.05 inches. In the sys- 

tern of teeth adopted by the Brown & Sharpe Mfg. Co., and 
which is used very extensively in America, the following pro- 
portions are given for cut teeth. See Fig. 151. 
Let D x — the outside diameter of the gear. 
" D = the pitch diameter of the gear. 
" D 2 = the diameter of a circle through bottom of space. 
" S = the diametral pitch. 
" s = the circular pitch. 

" a = the addendum = height of tooth above pitch line. 
" c = the clearance between top of tooth and bottom of 

space when gears are in mesh. 
" d = the dedendum, or total depth of space below pitch 

line. 
" t = the thickness of tooth on pitch line = width of 

space on pitch line in cut teeth. 
" N = the number of teeth in gear. 
" h = the total height of tooth. 

Then N = D S = — , 

s 

s n 
t = - = — 
2 26 

t_ 7t 

IO 20 S 



TOOTHED GEARING 369 

I 

d = a + c 

h = 2 a + c 

N + 2 
Z) x = — - — and D 2 = D — 2 (a + c) 

In the case of rough gear teeth, cast from a wooden pattern, the 
thickness of the tooth must be less than the width of the space,* 
and the clearance at the bottom of the space must be greater than 



in cut teeth. If the gears are machine-moulded, the difference 
need not be quite so great as in pattern-moulded gears. For 
pattern-moulded gears good practice gives t = 0.45 5 for large 
gears, to 0.47 5 for small gears, and the corresponding width of 
the space would be 0.55 5 to 0.53 s. For machine-moulded gears 
t = 0.465 to 0.48 ^ and the corresponding space would be 0.54 s 
to 0.52 s. 

Table XXVI gives dimensions of gear teeth for cut spur gears, 
in accordance with the standards of the Brown & Sharpe Mfg. Co. 

171. Methods of Making Gear Teeth. Metallic gear wheels 
are either cast from a pattern, or the rim is cast or forged solid, 
and the teeth are cut from the solid metal by rotary or recipro- 
cating cutters. Where the gear teeth are cast, it is very important 
that the pattern itself be very accurately rnade; for even with the 
greatest care in moulding, it is impossible to obtain true spacing, 
on account of shrinkage and displacement due to " rapping" the 
pattern in the sand. For this reason, and on account of the 
difficulty of obtaining smooth surfaces, greater clearance must be 
allowed in cast gears than in cut gears, as already noted. Wooden 
patterns are very unreliable for such work, on account of their 
tendency to warp and shrink, and permanent patterns should be 
made of metal. If the pattern for a spur gear is withdrawn from 
the sand with a movement parallel to the length of the tooth, the 
tooth pattern must have draft, or be slightly tapering to facilitate 
drawing, and consequently the cast tooth must also be tapering. 
Care should be taken in assembling such gears, that the tapers in 

* The difference between the thickness of the tooth and the width of the space 
is commonly called " backlash." 
24 



37° 



MACHINE DESIGN 



the two gears are reversed to avoid having the thick ends of 
both sets of teeth come together, thus concentrating the pressure 
at one end. Rough cast gears, of the kind described above, are 
used only for rough or large work, and not for high speed. The 
particular defect of spur gears due to draft does not exist in bevel 
gearing. 

In gear-moulding machines the pattern consists of a segment 
of the gear pattern, carrying several teeth. The pattern is 

table xxyi 

PROPORTIONS OF GEAR TEETH 



Diametral 


Circular 


Thickness of 


Addendum Dep 
1 Space 
~S =a - Pitch 


th of 
Below 
Line. 


Total Depth 


Pitch. 


Pitch. 


Tooth. 


of Tooth. 


S 


s 


t 


a a 


+ c 


2 a + c 


I 


3.1416 


1.5708 


I . OOOO 1 


1571 


2.1571 


ll 


2 -5 J 33 


1 .2566 


.8000 


9257 


I-7257 


I* 


2.0944 


1 .0472 


.6666 


7714 


i-438i 


If 


1-7952 


.8976 


•57M 


6612 


1 .2326 


2 


1.5708 


•7854 


.5000 


5785 


1.0785 


4 


1-3963 


.6981 


•4444 


5143 


•9587 


2§ 


1 .2566 


.6283 


.4000 


4628 


.8628 


2| 


1 .1424 


•5712 


■3 6 3 6 


4208 


.7844 


3 


1 .0472 


•5236 


•3333 


3857 


.7190 


3* 


.8976 


.4488 


.2857 


33° 6 


.6163 


4 


•7854 


•3927 


.2500 


2893 


•5393 


5 


.6283 


.3142 


.2000 


2314 


•43*4 


6 


•5 2 3 6 


.2618 


.1666 


1928 


•3595 


7 


.4488 


.2244 


.1429 


1653 


.3082 


8 


•39 2 7 


.1963 


.1250 


1446 


.2696 


9 


•3491 


•1745 


.1111 


1286 


•2397 


IO 


•3 J 42 


•1571 


.1000 


"57 


•2157 


ii 


.2856 


.1428 


.0909 


1052 


.1961 


12 


.2618 


.1309 


• o8 33 


0964 


.1798 


13 


.2417 


.1208 


.0769 


0890 


.1659 


14 


.2244 


.1122 


.0714 


0826 


•i54i 


15 


.2094 


.1047 


.0666 


0771 


•1437 


16 


.1963 


.0982 


.0625 


0723 


.1348 


i7 


. 1848 


.0924 


.0588 


0681 


.1269 


18 


•1745 


.0873 


•o555 


0643 


.1198 


19 


•1653 


.0827 


.0526 


0609 


•"35 


20 


•i57i 


•0785 


.0500 


o579 


.1079 



mounted on an axis in such a manner that it can be rotated 
accurately through any portion of a complete revolution, or 
"indexed." In forming the mould the segmental pattern is 
placed in position and sand is rammed around it. The pattern 
is then withdrawn radially and rotated to the next succeeding 



TOOTHED GEARING 



371 



position (the indexing device insuring accurate spacing), the 
operation being repeated till the whole circumference is moulded. 
The mould for the hub and arms is then completed, in large work 
this last being often accomplished by means of cores. If machine 
moulding is well done the results are far superior to those obtained 
by pattern moulding, and gears may be made that can be run at 
moderately high speeds. Obviously, however, all cast gears are 
much more inaccurate than cut gears, and the latter are preferable 
where high speeds and smoothness of action are required. 

Metallic gearing, even when accurately cut and aligned, is 
inclined to be very noisy when run at a peripheral speed of more 
than 1,200 feet per minute, especially if any appreciable "back- 
lash" exists. Relieving the points of the teeth, slightly, reduces 
the tendency to produce noise. Where high speeds are unavoid- 
able the teeth of one of the mating gears is sometimes made of 
wood or rawhide. Wheels with wooden teeth are known as 
mortise wheels. They are not as much used as formerly, because 
modern methods of gear-cutting produce metallic gears of such 
accurate form that they may be run in places where mortise gears 
were formerly considered indispensable. In making mortise 
wheels the wooden teeth are roughed out and the shank is fitted 
into openings cast in the rim of the wheel, as shown in Figs. 149 
and 150. The teeth are held in place by the keys, K, or pins, P, 





Fig. 150. 



as shown. The teeth proper are dressed to correct form with 
hand tools or by special machines using a fine circular saw for a 
cutter. 

Usually the large gear, only, is made with wooden or " mortise" 
teeth, the pinion being made of metal. This is rational since 
the pinion, on account of the shape of its teeth, is the weaker of 



37 2 MACHINE DESIGN 

the two, and also because the teeth of the pinion come into contact 
more frequently, and hence suffer greater wear. In such com- 
binations, the metal gear frequently has teeth of thickness 

less than - and the wooden gear teeth of thickness greater than 

-, to equalize strength. See Fig. 150. In recent years gears 

made of rawhide have been much used for high speeds. 
The blanks for rawhide gears are made by cementing specially 
prepared rawhide discs together under great pressure. Me- 
tallic discs, on each side of the blank, held together by rivets 
passing through the blank, assist the rawhide teeth in retaining 
their form. The teeth are cut in the blank in the same manner 
that metallic teeth are cut. In using rawhide gearing the 
pinion is almost always made of rawhide and the larger gear of 
cast iron or brass. Such a combination may be run at a very high 
rate of speed, 3,000 feet per minute being a not unusual velocity. 
Rawhide gears are almost noiseless in operation but care must be 
used that they are not subjected to extreme moisture nor run in 
too dry an atmosphere. 

Formerly it was cheaper to cast gear teeth, but the development 
of gear-cutting machinery has changed the situation where a large 
number of gears with small teeth are to be made. Modern 
methods of gear-cutting produce teeth of great accuracy, and 
have also so greatly reduced the cost of production that for high 
speeds, and where smoothness of action is necessary, cut gears 
have largely superseded cast gears even in large work. 

There are many methods of cutting gear teeth in practical 
operation, the most common method of cutting spur gears being 
by the use of a rotating cutter.* The outlines of gear teeth vary 
with the number of teeth in the gear, the pitch or thickness of 
tooth remaining constant, and, theoretically, a different cutter is 
required for every different diameter of gear in a series of the 
same pitch. To meet this requirement would lead to an excessive 
number of cutters for each pitch. It is found in practice, how- 
ever, that the same cutter can be used, without serious error, for 

* See " Gear-Cutting Machinery," by Ralph E. Flanders. 



TOOTHED GEARING 



373 



several sizes of gears of a given pitch. In the system adopted by 
the Brown & Sharpe Mfg. Co., only 24 cutters are used for each 
pitch in the cycloidal system, and only 8 cutters for each pitch in 
the involute system, as given below. The letters and numbers 
in the first column are the manufacturer's designations, for pur- 
poses of ordering cutters. 













TABLE 


XXVII 
















CUTTERS 


FOR 


CYCLOIDAL TEETH 






Cutter A 


cuts 


12 


teeth. 








Cutter M 


cuts 


27 to 


29 tee 


(< 


B 


tl 


r 3 


n 








a 


N 


a 


30 " 


33 


a 


C 


tt 


14 


n 








a 


O 


tt 


34 " 


37 


tt 


D 


tt 


15 


a 








a 


P 


a 


38 " 


42 


a 


E 


tt 


16 


it 








tt 


Q 


it 


43 " 


49 


tt 


F 


tt 


17 


ti 








ti 


R 


it 


5o " 


59 


tt 


G 


it 


18 


a 








a 


S 


it 


60 " 


74 


it 


H 


tt 


19 


a 








a 


T 


11 


75 " 


99 


it 


I 


tt 


20 


tt 








a 


U 


11 


100 " 


149 


a 


J 


it 


21 


tO 22 


teeth 






tt 


V 


it 


150 " 


249 


it 


K 


a 


2 3 


" 24 


a 






a 


w 


ti 


2^0 or 


more " 


a 


L 


it 


2S 


" 26 


n 






a 


X 


a 


Rack. 





TABLE XXVIII 

CUTTERS FOR INVOLUTE TEETH 

Cutter No. i cuts from 134 teeth to rack. 

2 " " 55 to 134 teeth. 

, 'i a -- a -. a 

3 3b 54 

4 " " 26 " 34 " 

5 "- " 21 " 25 " 

6 " " 17 " 20 " 

7 " " 14 " 16 " 

8 " " 12 " 13 " 

When gear-cutting is carefully done, very accurate work may 
be accomplished. It is to be noted, however, that the form of 
the teeth when cut with a set of cutters, as above, are not all 
theoretically correct;* and even in best practice the error in the 
gear-cutting machine itself, coupled with that due to dullness of 
cutters and deviation due to different degrees of hardness in the 
metal, may be considerable. 

172. Forces Acting on Spur Gears. In Fig. 151 let the gear A 
drive the gear B. Let F a be the velocity of the pitch circle of A; 
and V h be the velocity of the pitch circle of B. Also let W & be 
the equivalent driving force acting at the pitch circle of A , and 
let W b be the equivalent resisting force acting at the pitch circle 

* There are gear-cutting machines which, theoretically, generate correct forms 
of teeth for all gears of a series. 



374 



MACHINE DESIGN 



of B. If now the tooth outlines are properly constructed, the line 
of action of the actual driving force W 1 will always pass through 
the pitch point and the angular velocity ratio of A to B will be 
constant. The action of the pitch circles will be as though they 
rolled upon each other and their linear velocity will be the same 
or F a = V b . From the principle of work 

W V =W h V h Therefore W = W, 

a a b b a b 

The tangential driving force exerted by one gear upon another 
is, therefore, independent of the angle of pressure, in any correct 




Fig. 151. 



Fig. 153. 



system of gearing, and the action is, in this respect, the same as 
if a pair of teeth were continually in action at the pitch point. 

The distribution of the reaction at the bearings due to the 
pressure between teeth (W lf Fig. 151), and its bending effect on 
the shaft which supports B, will depend upon the relative positions 
of the gear and bearings; but the latter will, in any case, be 
directly proportional to W x . As the obliquity of the line of 
action C D is increased, the angle (Fig. 151) is increased and 
hence sec 6 is also increased. Therefore, since W x — W :i sec 9, 



TOOTHED GEARING 375 

the pressure on the bearings is increased with an increase of the 
obliquity of the line of action; but the torque on the driven shaft 
remains unchanged. 

In cycloidal gearing the obliquity varies from a maximum at 
the beginning of the contact to zero when the contact point lies 
in the line of centres; and, during the arc of recess, it increases 
to a maximum at the end of contact. The maximum value of 
the angle 0, Fig. 148, is about 22 , with usual forms of cycloidal 
teeth. When equals 22 , sec 6 equals 1.08, or the maximum 
normal pressure is about 8 per cent greater than the tangential 
rotative force. 

The obliquity is constant throughout the arc of action in 
involute gears, and the angle 0, Fig. 147, is usually 14K or 1 S°- 
When = 15 , sec = 1.035, or the normal pressure is 3X per 
cent greater than the tangential force. In the above discussion 
the influence of friction has been neglected. During the arc of 
approach the frictional force F (Fig. 151) deflects the line of 
action of W 1 in such a way as to increase the effective obliquity. 
During the arc of recess it acts in the opposite direction and de- 
creases the obliquity. The influence of this frictional force is 
small and may, usually, be neglected, but its action accounts, 
to a certain degree, for the well-known fact that gears run 
more smoothly during recess than during approach. 

It is usually intended that more than one pair of teeth shall 
be in action at all times, but, owing to the unavoidable inaccuracy 
of form and spacing previously noted, it is not safe to depend upon 
a distribution of the load between two or more teeth of a gear. It 
is safest to provide sufficient strength for carrying the entire load 
on a single tooth. In the rougher classes of work, this load may 
be concentrated at one end of the tooth, as indicated in Fig. 152, 
and all such gears should be carefully inspected and corrected, 
if intended to carry heavy and important loads. With well 
supported bearings, and machine-moulded or cut gears, it is not 
unreasonable to consider the load as fairly well distributed across 
the face of the gear, if the face does not exceed in width about 
three times the circular pitch (see Fig. 153). 

The obliquity of the line of pressure gives rise to a crushing 



376 



MACHINE DESIGN 



action on the teeth (due to the radial component of the normal 
force), in addition to the flexural stress which results from the 
tangential component. This crushing component, with the 
ordinary proportions of teeth, does not exceed 10 per cent of the 
normal pressure. Its effect is to reduce the tensile stress due to 
flexure, and to increase the compressive stress. Since cast iron 
is far stronger in compression than in tension, this may be 
neglected in gears made of that metal, while in the case of steel, 
or composition gears, the margin of safety assumed usually makes 
it unnecessary to consider this component. 

173. Strength of Spur Gear Teeth. The assumption often 
made that the teeth of spur gears can be considered as rectangular 
cantilevers, in determining their strength, is not satisfactory, es- 
pecially when treating pinions having a small number of teeth. Fig. 
154 shows four gear teeth which have the same thickness at the 





pitch line and the same height. The tooth marked (a) is one of 
an involute rack; (b) is one of an involute pinion having 12 
teeth; * (c) is one of a cycloidal gear having 30 teeth; (d) is one 
of a cycloidal pinion of 12 teeth. 

Mr. Wilfred Lewis, of Wm. Sellers & Co., seems to have been 
the first to investigate the strength of gear teeth with due regard 
to the actual forms used in modern gearing. His work was 
published originally in the Proceedings of the Engineers' Club of 
Philadelphia, in January, 1893, and his method of investigation 
was as follows : Accurate drawings of gear teeth were made on a 
large scale, and the line of action of the normal force, when acting 
on the point of a tooth was drawn in; see Fig. 155. From the 

* The 12-tooth involute pinion may have its teeth weakened by a correction 
for interference; but it is usually better to correct the points of the mating gear. 



TOOTHED GEARING 



377 



intersection of this line of action with the centre line of the tooth, 
a parabola was drawn tangent to the sides of the tooth, thus 
locating a beam of uniform strength equal to the effective strength 
of the tooth (see Article 15). The points of tangency a, a, locate 
the weakest section of the tooth, and the bending moment applied 
to this section is W I. Then from equation J, page 94. 

= y 3 t>ph 2 = bps(y 3 -) 



Wl 



I pb(2h 
h 2 \ 



orW = bps [ 2 / 3 —J = bps (y) ' (1) 

Where b = the breadth of the tooth in inches, p = the tensile 
stress, and s = the circular pitch. The factor y is a variable, 
depending on the shape of the tooth. Mr. 
Lewis found that its value is practically 
independent of the pitch (since s, h and / 
are proportional to the pitch), but depend- 
ent mainly on the number of teeth in the 
gear. Tabulated values of this coefficient may 
be found in Kent's " Mechanical Engineers' 
Pocketbook," page 901. From these tabu- 
lated values, Mr. Lewis deduced the follow- 
ing equations in which N = the number of teeth in the gear. 

For the 15 involute system and the cycloidal system using 
a generating circle whose diameter equals the radius of the 12- 
tooth pinion, 

„• . / o.684\ 
W = bps ^0.124 —J 




Fig. 155. 



(2) 



For the 20 involute system, 

W = b p s (.154 



912 

7^ 



(3) 



Mr. Lewis' investigations on cycloidal gears were made on a 
system using the radius of the 12-tooth pinion as the diameter 
of the describing circle. Modern practice sometimes makes the 
radius of the 15-tooth pinion the diameter of the describing circle, 
which gives somewhat weaker teeth than the first system. The 



378 MACHINE DESIGN 

difference is small, however, compared to the variation in the 
assumed stress, p, and since cycloidal teeth are now little used 
for small and moderate-sized gears, equation (2) will be adopted 
in this work for standard gears. 

The Lewis' formula is convenient for determining W, b, s, or p, 
where the number of teeth (N) is known; but a very common 
problem in design is to determine the pitch (s), when the pitch 
diameter of the gear is given and the number of the teeth is un- 
known. The formula may be adapted to this last stated problem 
as follows.* To accord with modern practice, circular pitch will 
also be transformed to diametral pitch. 

Let D = the pitch diameter. 
" w = the load per inch of face. 

" S =- the diametral pitch = — or s = -^ 

Then N = D X S * 

/ Ma\ tz / .684\ 

Therefore W = bsp ^.124 - — -J = b X ^ X p ^.124- --) 

«?->*(f - a S « 

or since w = W + b 

"><&-$) ■ ■ ■ ■ « 

and therefore S = *-(. 194 + J Q .g _^1S^\ ... (6) 
w \ \ ' ° p D ' 

The pitch can be found from equation (6) for any values of w, 
D, and p, when the face of the gear is known or assumed. A 
common problem is as follows: The distance between two 
shafts and their velocity ratio is known; required the pitch of spur 
gears to connect these shafts for a given load and working stress 
on the teeth. The centre distance of the shafts, and the velocity 
ratio fix the diameter of the gears. The face of the gears may 
be governed by the space available, or it may be assumed by the 
designer upon other considerations. To illustrate; suppose 

* See a discussion by John H. Barr r Trans. A. S. M. E., Vol. XVIII, page 766. 



TOOTHED GEARING 379 

W = 15,000 lbs., p = 8,000 lbs. per square inch, and that the 
smaller gear is to be 40 inches diameter. Assume also that the 
face of the gear may be taken as 6 inches. The load per inch of 
face is w = 15,000 -f- 6 = 2,500 lbs., hence, 

8,000/ I 2. is X 2,c;oo\ 

s = -i— .194 + J .038 — ^ — ) = I - 1 or sa y x 

2,500 V > ° 8,000x40 / 

diametral pitch. 

The diagrams shown in Figs. 156 and 157 are plotted from 
equation (5). That in Fig. 156 covers the range from 12 to 6 
diametral pitch and Fig. 157 covers the range from 5 to 1 diametral 
pitch. The abscissas (Scale A) represent pitch diameters of gears 
in inches, and the ordinates (Scale B) the load in pounds per inch 
of width of face, for a stress of 6,000 pounds per square inch. Any 
other stress could have been taken for plotting the diagrams, and 
any other may be used in solving problems by them. A curve 
is drawn for each pitch; to illustrate, let S = 1.5, let p = 6,000. 
Substituting these values in (5) 



7.389 2.15 x 

w = 6,000 ( — — ) 

V 1.5 D x 2.25; 

/ Tr ^ 5,73° \ 



or w = 

hence when D = 3.7, w = o; D = io, w = 983; D — 20, 
w = 1,270, etc. 

Plotting these corresponding values of D and w as abscissas 
and ordinates, respectively, the curve for a diametral pitch = 1% 
is drawn through the points. The other curves are constructed 
in a similar manner. If, then, the diameter of the gear is known, 
the allowable load per inch of face for a stress of 6,000 pounds 
per square inch may be found by passing vertically upward from 
the given diameter on scale A, to the curve corresponding to the 
pitch, and then moving horizontally to the left-hand scale B, 
which gives the required load per inch of face. Scale B is re- 
produced at the top of the diagram, as scale C, and a 45 diagonal 
marked 6,000 is drawn from the lower right-hand corner of the 



3 8o 



MACHINE DESIGN 



diagram to scale C. If, then, instead of moving horizontally 
from the pitch curve to scale B on the left, the movement be 



Scale B 



g 8? & 

o o o 



O 



ON 




horizontally ,to the right (or left) to the diagonal marked 6,ooo, 
and then vertically upward to the scale C, the same reading will 
be obtained on C as originally found on B. 



TOOTHED GEARING 



38l 



Furthermore, if other diagonals be drawn, as shown, from 
various points on scale C, they may be used to read loads per inch 



Scale, B. 







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Diametral Pitch 


- H*£ « £ » £ 


h> 






Circular Pitch 


2.09 

1.57 

1.27 

1.05 

.897 

.785 


CO 

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of face for stresses corresponding to these respective points on 
this scale, since from equation (4) it appears that the stress 
varies directly with the load. These stresses are indicated along 



382 MACHINE DESIGN 

the several diagonals. Thus to find the pitch when w = 2,500, 
P = 8,000, D = 40, from 2,500 on scale (C), Fig. 157, pass verti- 
cally downward to the diagonal marked p = 8,000; then horizon- 
tally to a point on the vertical rising from D = 40", on scale A. 
The pitch curve nearest this point is for 1 diametral pitch which 
would be the required pitch. 

If it is required to find the load per inch of face for a gear of 
given diameter and pitch, with an assigned stress, start at the 
point on scale (A), corresponding to the diameter; pass upward to 
the given pitch curve; thence horizontally (right or left as the case 
may be), to the proper stress diagonal; thence upward to scale 
(C), where the unit load may be read off. (See Fig. 156 or 157.) 

If the diameter, pitch, and unit load are known quantities, 
pass upward from the diameter reading on scale A, to the proper 
pitch curve; thence horizontally to a point under the unit load 
on scale C, and the stress may be found by interpolating between 
the adjoining diagonals. 

It may be noticed that the different pitch curves in either Fig. 
156 or 157 have a common tangent through the origin O. The 
points of tangency correspond to the diameters of gear at which 
cycloidal teeth have radial flanks for their respective pitches; i.e., 
a 12-tooth gear. The various curves have not been extended far 
beyond this point as in that case they intersect, and mar the 
clearness of the diagram. Since this intersection occurs after 
the diameter of the 1 2-tooth gear is reached, it is evident that the 
remainder of the curve is of no practical importance. In fact, 
increase of pitch, or a less number of teeth for a given diameter, 
beyond this point will not give additional strength, because the 
length of tooth will be increased, and the flanks will be under- 
cut to an extent which more than compensates for the added 
thickness of tooth. 

The data may be such that a point corresponding to a, Fig. 
157, will lie to the left and above all the pitch curves, i.e., above 
the common tangent through O. If the same data were substituted 
in equation (4) an imaginary quantity would result. This means 
that the unit load taken cannot be carried with the stress and 
diameter assumed, by any possible pitch. The only recourse for 



TOOTHED GEARING 



383 



a gear, of the given diameter and total load W, is to increase the 
face, and thus reduce w, or to use a material which permits a 
higher intensity of stress. 

It should be noted that the teeth of the smaller gear of a mating 
pair are weaker in form than those of the larger. The wear, also, 
is greater on the teeth of the smaller gear since they come in con- 
tact more frequently. Hence, in general, if the small gear is 
properly designed the larger gear will have sufficient strength. 




Fig. 158. 



This does not apply to certain forms of reinforced or " shrouded" 
teeth discussed later, nor, necessarily, where the thickness of 
teeth and spaces are unequal, nor where the mating gears are of 
different material. 

174. Strength of Bevel Gear Teeth. If a pair of bevel gear 
teeth, Fig. 158, have just come into contact as shown at a, Fig. 
147, then the driving force is applied to the point of the driven 
tooth by the root of the driver. The tooth of the driven wheel 
will be deflected a certain amount, while the deflection of the driv- 
ing tooth will be negligible. Since the deflection of the driven tooth 
is caused by the rotative effort of the driving gear, the magnitude 



384 MACHINE DESIGN 

of this deflection at any point on the line of contact of the two 
teeth will be proportional to the movement of the corresponding 
contact point of the driver or to its distance from the axis of 
rotation of the driver; hence, from similar triangles, will also be 
proportional to the distance from its own axis of rotation. Now 
the cross-sectional outlines of the tooth are similar at all points, 
and it can be shown that in the case of simple cantilevers of 
similar form the load applied is proportional to deflection. It 
has just been shown, however, that the deflection of the tooth at 
any point is proportional to the distance of that point from the 
axis of rotation. Hence the load on the tooth at any point must 
also be proportional to the distance from the axis, being least at 
the small end, and greatest at the large end, the mean value be- 
ing at the middle of the tooth. Therefore a spur gear which has 
the same width of face, and teeth of the same form and pitch 
as the mean section, will have, theoretically, the same strength as 
the bevel gear. It can also be shown that in simple cantilevers 
of equal breadth and similar outline, the stresses induced at cor- 
responding points on the cantilevers are equal, if the load 
applied is proportional to the linear dimensions. Hence the 
maximum stresses are the same at all sections of the tooth. 

It is evident that the relation established above between the 
mean section of a bevel gear and a spur gear with similar teeth, 
may be (and often is) used as a means of designing bevel gears. 
It is much more convenient, however, to deal with the teeth at 
the outer or large end. If, also, the pitch radii are used instead 
of the addendum radii, the error will not be great. 

Let r x = the pitch radius at the small end of the tooth. 

r 2 = the pitch radius at the large end of the tooth. 

r = the mean pitch radius of the tooth. 

b = the width of the face of the tooth along the elements. 

w = the load per inch of face at the radius r. 

w 2 = the load per inch of face at the radius r 2 . 

W = the resultant load on the tooth = w b. 

w e = the equivalent load per inch of face which, if acting 
at a radius r 2 , would produce the same rotative effect as the 
actual load. 



TOOTHED GEARING 385 

Since the load on the tooth varies as the radii, the total re- 

2 (V- 3 — Y 3 ) 

sultant load will act at a radius R = — -— ~, and the torsional 

3 (>V — O 

moment due to the resultant force, W, will be 

WR = -wb i )\~ r ^ . 
3 (^ 2 — r?) 

Now by definition w e b r 2 = W R. 

2 (y 3 — ^ 3 ^ 
Therefore, w = —w —^ — * (7) 

Also, since the load varies with the radius, 

W 2 W ' W 2W 2WY 2 

— = - = ; = . * . W 2 = ... (8) 

Y 2 y y 2 + Y t r 2 + r t y 2 + r t 



And from (7) and (8) 

7^2 = 2W y 2 ^ l w (r 2 * — ^i 3 ) __ 3 ( r 2 ~ O r 2 2 * 

w e " " r 2 + r t 3 W (r 2 2 — y x 2 ) y 2 " (r 2 3 — r t 3 ) {9) 

The actual load, w 2 , will always be greater than w t in the ratio 
shown above. A bevel gear, therefore, will be more heavil}' 
loaded at the large end than a spur gear of the same diameter, and 
carrying the same torque, in the ratio shown above. If, however. 
w e is known, w 2 can be computed, and used in equations (5) and 
(6) of Art. 173 instead of w. Usually r t is made not less thar 

2 2 w 2 

—y 2 . When r. = — r 2 , — = k = 1.4, and this value can be used 

3 3 ^ 

in computing w 2 unless the face of the gear is excessively long. 
It should be especially noted that in solving problems in bevel 
gearing, either by the diagrams of Figs. 156 and 157 or by 
equations (5) and (6) of Art. 173, the diameter Z>, which must 
be substituted therein, is that corresponding to the foYmative 
ctYcle, whose radius is R f = r 2 sec 6, Fig. 158, as the foYtn of 
the tooth is fixed by this radius and not by the radius y 2 . The 

* See also Mr. Lewis's article, Proceedings Engineers' Club of Philadelphia, 
Jan., 1893. 
25 



386 MACHINE DESIGN 

computations should be made for the smaller of the two gears, as 
in the case of spur gears. 

Example. Design a pair of bevel gears to transmit 50 H.-P. 
with a velocity ratio of 3 to 2; the gears to be of cast iron, and 
the maximum fibre stress to be 4,000 pounds per square inch. 
The revolutions per minute of the shafts are to be 300 and 200 
respectively. 

Lay off the axes OV and OT, Fig. 158, and draw OTJ so 
that corresponding radii of N and M are in the proportion of 3 
to 2. Then it is found that 6 = 34 and 0' = 56 . Assume tenta- 
tively that r 2 for the large gear = 15" and r 2 for the small gear 
= 10", and take a trial width of face of 4". 

_. t . . .. 2 X 7T X IO X 300 

The velocity at the radius r 2 = = 1,575 

feet per minute. Hence the equivalent total load at a radius r 2 

50 X 33,000 1,050 
= W = — = 1.050, or w = = 262 pounds. 

Therefore w 2 = w e X k = 262 X 1.4 = 367 pounds per inch of 
face. Since sec = sec 34 = 1.2, the diameter of the formative 
circle is 20 X 1.2 = 24" and from the diagram, Fig. 157, or from 
equation (6), it is found that for D = 24", p = 4,000 pounds, 
and w = 367 pounds, the diametral pitch is very nearly 3%, 
which may therefore be selected. This would give 70 teeth for 
the small gear and 105 for the large gear. The width of face is a 
little more than three times the circular pitch and is therefore in 
accordance with good practice. 

175. Stresses in Gear Teeth. Gear teeth of all kinds are likely 
to be subjected to shock, unless running at a very low velocity, 
and the danger from shock increases as the velocity increases; 
hence the allowable stress must be reduced as the velocity is 
increased. Reliable experimental data on the allowable stress 
in gear teeth are lacking, although many empirical rules are to 
be found in treatises on the subject. The values given by Mr. 
Lewis, in the paper already quoted, are probably as reliable as 
any, for teeth that bear along their entire length. The following 
equations have been deduced from Lewis's work: 



TOOTHED GEARING 387 

/ 600 \ 

For cast iron, p = 8,000 f 7 — ) . . (10) 

r \6oo + VJ 

/ 600 \ 

for steel p = 20,000 ( ) . . (11) 

r V600 + VJ 

where V = the velocity at pitch line in feet per minute. It would 

be probably safe to take the stress 

/ 600 \ 

for bronze as p = 12,000 (- — ) . . (12) 

r \6oo + VJ 

since the resilience of bronze is greater than that of cast iron. 
An old empirical rule for rough cast teeth is 

W = 200 X s X b (13) 

where W, as before, is the total load, s the circular pitch, and b 
the width of face of tooth. The strength of wooden mortise teeth, 
made of beech or maple, may be taken as about one-half that of 
cast iron, under the same circumstances; and the strength of 
good rawhide gears may be taken as equal to that of similar gears 
made of cast iron. It is to be noted that a rawhide gear will en- 
dure considerable more shock than one made of cast iron. 

While rough cast teeth are more likely to bear on one corner 
only, they are stronger than cut teeth of the same pitch, 
which compensates in a measure for this defect; furthermore, 
there is, usually, an excess of strength, to allow for wear, in all 
new gears, and the subsequent wear tends to correct the initial 
unequal bearing, along the elements. 

On account of the increased liability to shock, with increase 
of speed, and also because of the noise of operation at high speeds, 
there is a limit to the speed at which any form of gear may be 
safely and conveniently operated. Mr. A. Fowler, in Engineering, 
April, 1889, gives the following as maxrmum values at which 
gearing may be successfully operated: 

Ft. per Min. 

Ordinary cast-iron gears 1,800 

Helical cast-iron gears 2,400 

Mortise wheel and cast-iron pinion 2,400 

Ordinary cast-steel gears 2,400 

Helical cast-steel gears 3>ooo 

Special cast-iron machine-cut gears 3,000 



388 MACHINE DESIGN 

Although higher velocities are occasionally found in practice, 
these are undoubtedly maximum average values and, in general, 
the velocity should not be more than two-thirds the values given 
above, on account of noise and wear. Rawhide gearing, which 
operates almost noiselessly, may be run satisfactorily up to 3,000 
feet per minute. 

176. Width of Face of Gears. Equation (5) of Art. 173 gives 
the load per inch of face that may be applied to a tooth of the 
given form and pitch, the total load depending on the width of 
face as shown by equation (4) of the same article. The dur- 
ability of a tooth for a given load is, therefore, theoretically 
increased with increase of face. The difficulty of securing 
uniform distribution of the load along the contact element in- 
creases, however, as the width of face is increased, and this imposes 
a practical limit to the width of the face. On the other hand, if the 
intensity of the load on the tooth is too great, excessive wear 
may result. The equations given above do not take wear into 
account, the allowable load being fixed with reference to the 
stress alone. On this basis a large tooth may carry a much 
higher load per inch of face, but the wear will be proportion- 
ally greater, the velocity being the same. The empirical rule 
given in equation (13) of Art. 175 assigns a load of 200 pounds 
per inch of face, per inch of circular pitch. For a tooth of 1" 
circular pitch this load will give, by the Lewis equation, a stress 
of only 2,000 pounds per square inch, for moderate -sized gears. 
This is a very low stress, for ordinary speeds, so that this rule 
would give more durable teeth than the Lewis equation, as 
ordinarily applied. 

Experimental data on the durability of teeth are lacking. It 
is evident, however, that the allowable load will depend largely 
on the character of the service, velocity of rubbing, lubrication, 
and the material used. Thus, for ordinary cut cast-iron teeth 
under constant service, the value given above (200 lbs.) is prob- 
ably conservative; while with teeth of high-grade steel much 
greater loads may be carried. Cases are on record where loads 
of over 2,000 pounds per inch of face were, successfully carried, 
with a peripheral velocity of over 2,000 feet per minute, the 



TOOTHED GEARING 389 

pinion being of forged steel and the gear a steel casting, 4.92" 
circular pitch. Well-made gears of rawhide may be loaded up 
to 150 pounds per inch of face, per inch of circular pitch; but in 
no case should the load exceed 250 pounds per inch of face.* 
In the case of machines such as punching-machines which 
work intermittently, and whose operation extends over a short 
space of time, the element of wear is not so important in the de- 
sign of the teeth; but in such gears as those connecting street- 
railway or automobile motors with the driving axles, where the 
work is both continuous and severe, wearing qualities may be 
fully as important as strength; and gears made of steel or other 
hard materials may have to be used solely on this account. 

Good practice makes the face of the tooth about three times 
the circular pitch ; but in fixing the pitch and width of face, in 
extreme cases, the points discussed above should be con- 
sidered. 

177. Other Forms of Gear Teeth. Gear teeth made according 
to the Brown & Sharpe standard, on which the foregoing dis- 
cussion is based, have been found very satisfactory for average 
conditions, and are in most common use in this country. For 
extreme conditions, however, it has been found necessary to 
reinforce such teeth, or to use teeth of a different form. 

A very common way of reinforcing teeth of cast gears is by 
shrouding, which consists in casting an annular ring of metal on 
one or both ends of the teeth, as shown in Fig. 159. This ring is 
cast as an integral part of the gear casting, and hence strengthens 
the gear tooth by practically twice the shearing strength of the 
cross-section of the tooth, when both ends are shrouded to the 
top. The teeth of the pinion are, from their outline, always 
weaker than those of the gear, and the wear on them is also 
greatest. The shrouding should, therefore, be put on the pinion; 
and if carried to the top of the tooth on both ends it will give 
them an excess of strength over those of the gear, with usual 
widths of face. If the gears to be reinforced do not differ greatly 
in diameter, the teeth of both may be shrouded half way up. 

* Private communication from the New Process Raw Hide Co. 



390 MACHINE DESIGN 

Shrouding is used mostly on rough cast gears, the shroud practi- 
cally prohibiting the cutting of the teeth by the usual methods. 

If the gears are to run in one direction only, and where very 
heavy pressures are to be withstood, a form of tooth as shown in 
Fig. 1 60, and known as a buttress tooth, may be, but seldom is, 
employed. The driving face, A, is made of correct theoretical 
outline, while the back face B may be of any outline* that will 
give the required strength, and clear the teeth of the mating gear. 
The front face should be of standard cycloidal or involute form 
and the backs are preferably involute forms, with a much greater 
obliquity of generator than would be permissible in driving. 

For some time past there has been a marked tendency f on 
the part of the designers of gearing for extremely trying service 
to depart from the Brown & Sharpe standard, and to use teeth 
somewhat shorter than those given by that standard. In some 
instances the same angle of pressure has been retained, while in 
others this angle has been increased. Mr. C. W. Hunt reported 
to the A. S. M. E. in 1897 (Vol. XVIII) the results of the adop- 
tion of such a system and gives full information for their design. 
A few other manufacturers have adopted similar systems. The 
need of a small gear of great strength, in automobile work, has 
increased the demand for a stronger form of tooth, and it would 
seem that the old standard must be modified or a second standard 
adopted for extreme service. The most prominent form of these 
so-called " stub teeth," at present, is that advocated by the 
Fellows Gear Shaper Co. In this system an involute tooth 
with a pressure angle of 20 is used, the addendum being about 
0.8 as high as that of the Brown & Sharpe standard. This 
gives a tooth nearly twice as strong as the old standard. Some- 
times these stub teeth are given the height of a standard tooth of 
smaller pitch; thus a 6-pitch stub tooth may have the length of 
a standard 8-pitch tooth, in which case the gear is sometimes 
described as a 6-8 gear. Notwithstanding the fact that the arc 
of contact in stub-tooth gears is, generally speaking, less than in 

* See " Kinematics of Machinery," by John H. Barr, page 131. 
f See a paper by R. E. Flanders, Trans. A. S. M. E., Vol. XXX, resume of 
other systems 



TOOTHED GEARING 



391 



the old standard, they run well and will undoubtedly be more 
used in the future. 

178. Strength of Gear Rims and Arms. The rim of the gear 
wheel must not only be strong enough to resist the forces brought 
upon it, but stiff enough also to prevent improper action of the 
teeth due to springing of the rim. A section of rim between two 
arms may be considered as a beam fixed at the ends and carrying 
a load at the middle, the value of which is W 1 sin 6, Fig. 151. 
Good practice makes the thickness of the rim at least 1.25 t, 
where t is the thickness of the tooth on the pitch line. For small 
gears this proportion gives ample stiffness, but for very large 
gears stiffening ribs are also sometimes necessary. In many 
cases the thickness should be sufficient to allow of dovetail- 



Shroud 




Fig. 159. 




Fig. 160. 



Fig. 161. 



ing a tooth into the rim, in case of accidental breakage of one 
or more teeth. Gear wheels are seldom run at peripheral 
velocities which induce dangerous centrifugal stresses. The 
principles governing the design of such wheels are discussed, 
however, in Chap. XV. 

The arms of gear wheels may be treated as cantilevers, assum- 

W 

ing that each arm carries a load — , where n is the number of 

n 

arms, and W the tangential load. Computations for strength 
of either arms or rims must, however, be considered as giving 
minimum dimensions, stiffness being the prime requirement, and 
due regard must be paid to proportions of rim, arms, and hub, 
to minimize shrinkage stresses due to cooling. 



392 



MACHINE DESIGN 



179. Efficiency of Spur Gearing. The experimental data on 
the efficiency of spur gearing are very meagre. Probably the best 
available data are those obtained by Mr. Wilfred Lewis, for details 
of which see Trans. A. S. M. E., Vol. VII. His investigation 
was made with a cut spur pinion of 12 teeth meshing with a gear 
of 39 teeth. The circular pitch was i>£ inches and the face 3^ 
inches. The load varied from 430 pounds to 2,500 pounds per 
tooth, and the peripheral speed ranged from 3 feet to 200 feet 
per minute. The measurements included the friction at the 
teeth, and the friction at the bearings. The efficiency, as ob- 
served, varied from 90 per cent at a velocity of 3 feet per minute 
to over 98 per cent at 200 feet per minute. It appears that the 
friction at the teeth is a small part of the loss with good cut gears, 
the greater portion of the loss being at the journals. The effi- 
ciency of bevel gears is somewhat less than that of spur gears, 
on account of the axial thrust, which induces friction between 
the hub of the gear and the collar at the supporting bearing. 



HELICAL OR TWISTED GEARING 

180. General Principles. Suppose a spur gear to be cut into 
n small sections by a series of planes perpendicular to the axis of 
rotation. If each section be then placed a proper distance ahead 
or behind the adjacent section, Fig. 161 (a), it is evident that they 
may be so arranged that some one section is just coming into 
contact with its mating section when the n th section in advance 
of it is in contact at the pitch point. With such an arrangement 
some section will always be in contact near the pitch point, and 
there will always be approximately n points of contact with the 
mating gear between the pitch point and the point which marks 
the beginning of topth action. Since the action of gear teeth is 
smoothest when contact is near the pitch point, this arrangement 
of gearing runs more quietly and smoothly than ordinary spur 
gearing, and it was at one time used in machine tool and similar 
work where smooth action is very desirable. 

As the number of sections is increased, the total width of 
the gear remaining the same, the spacing of these sections being 



TOOTHED GEARING 



393 



kept uniform as before, the form of the stepped tooth approaches 
that shown in Fig. 161 (c). When the number becomes infinite 
the teeth become helical in form, and contact is continuous along 
that portion of the face which is within the arc of contact. It is 
evident, however, that since the relative position of adjacent 
laminae is arbitrary, and may follow any desired law, the outline of 
the tooth in an axial direction is not necessarily helical, but may 
have any desired shape; although these teeth are most usually 
made helical, this form being more practical to cut. This 
form of gearing is also known as twisted gearing, for an ob- 
vious reason. The action of such gears is identical with that 
of common spur gearing, and should not be confused with that 
of screw gearing, though certain limiting forms of the latter are 
also twisted gears. A screw gear must have regular or uni- 
form helical teeth, while a twisted gear does not necessarily 
have this limitation. 

Since the pressure, W, between mating teeth must be normal 
to the surface, there is a component, Fig. 161 (c), which tends to 
move the gear in an axial direction, causing end thrust on the 
shaft collars. This can be obviated by making tw r o sets of helical 
teeth on each gear, one right-hand and one left-hand, as shown in 
Fig. 162. When it is desired to use cut teeth the wheel is some- 
times made in two parts and fastened together, or the wheel may 
be made in one piece and the two sets of teeth staggered so as to 
allow them to be cut; but in both of these constructions there is 
some loss of strength due to the absence of the reinforcing action 
of teeth cast solid as in Fig. 162. Gears of this type are also 
called herring-bone gears. With the arrangement shown in Fig. 
162, care must be used that the alignment in an axial direction is 
accurate, or end play must be provided so that the middle plane 
of both gears coincide; otherwise the full load will be thrown 
on one-half the gear and the object of the double gear defeated. 

181. Strength of Twisted Gears. If the effective load which 
one tooth of a twisted gear transmits to its mate be W, Fig. 161 (c), 
then the total load normal to the face is W 1 = W cosec 9. If 
the length of the tooth be denoted by /, and the breadth of the gear 
by b, then / = b cosec 0. Hence the load per inch of face on 



394 



MACHINE DESIGN 



. . , W t WcosecO W 

a twisted tooth = — = — = -7- or the same as in a 

/ cosec 

spur gear of face b. This would be strictly true if all points in 
the li le of contact were at the same distance from the axis of ro- 
tation as in a spur gear. This is never so, in twisted gears, the line 
of contact always extending diagonally across the tooth face. 
The error due to this, however, is small, and on the side of safety, 
and it may be assumed that the load per inch of face in 
twisted gears is the same as that of a spur gear of equal width and 
equally loaded. This diagonal distribution of the load across 
the tooth face, decreases the lever arm of the force which 
tends to break the tooth; the amount of decrease depending 



^ 





Fig. 162. 



Fig. 163. 



Fig. 164. 



on the amount of twist in the tooth. If the twist is so great 
that when the end in advance is going out of contact the other end 
is just coming into contact, the line of contact will run diagonally 
across the tooth from point to flank, and the average arm of the 
driving force will be about one-half the height of the tooth. If 
the twist be made equal to the pitch, tooth action is continuous 
at every point of the arc of action and this proportion is the one 
most used. It is clear, however, that the assumption often made 
that twisted teeth are twice as strong as spur teeth of the same 
pitch is not true for teeth of usual proportions, a difference of 25 
per cent being, perhaps, as much as can safely be assumed. On 
account of continuous tooth action and consequent smoother 
operation in twisted gears, the effect of shock is lessened some- 



TOOTHED GEARING 395 

what. Twisted gears have been used with success on heavy wind- 
ing and hoisting engines, the teeth being often rough cast and 
both gear and pinion half shrouded, making a very strong tooth. 

SCREW GEARING 

182. Forms of Screw Gears. When the axis of two shafts are 
not parallel and do not intersect, it is possible to lay out contact 
surfaces on which gear teeth may be constructed which will give 
line contact. Gears of this kind are known as skew-bevel gears. 
They are difficult to construct, and are very rarely used. If 
the load can be carried on point contact, pitch cylinders may be 
described on the axes, Fig. 163, and on these surfaces helical 
teeth may be constructed which will transmit the desired motion. 
Such gears are known as screw or spiral* gears, the latter name 
being really a misnomer. While the teeth of such gears resemble 
those of helical twisted gears, their theory and action are quite 
different; for, in addition to the conjugate rolling and sliding 
action, as in spur gears, there is also a sliding component along 
the elements between contact surfaces. The action of screw 
gearing is very smooth. The special case where the axes are at 
right angles, and where a large wheel having many helical teeth 
meshes with a small one having a very few helical teeth, is an im- 
portant one on account of the great reduction in velocity ratio 
that may thus be obtained. This last arrangement is commonly 
known as a worm and worm-wheel. Fig. 165 illustrates such 
a worm and worm-wheel, the teeth on the worm wheel being 
truly helical in form and cut at an angle to suit the worm thread 
or helix. The same result is sometimes obtained by using a plain 
Spur gear, and setting the axis of the worm at the proper angle 
with the plane of the gear.| The contact in these cases is point 
contact, and on the worm wheel tooth is confined to points in a 
line cut from the working surface of the tooth by a plane passing 

* For a full discussion of the methods of laying out and producing so-called 
spiral gears, see a " Practical Treatise on Gearing," by Brown & Sharpe Mfg. Co., 
and also "Worm and Spiral Gearing," by F. A. Halsey. 

f A highly successful form of this arrangement is the worm-and-rack drives on 
planing machines, first used by Wm. Sellers & Co. 



39 6 



MACHINE DESIGN 



through the axis of the worm at right angles to the axis of the 
worm wheel. In practice the point of contact becomes a limited 
area. The advantage of this form of worm wheel, like all spur 
gears, is that the teeth can be cut with a rotary cutter, and patterns 
for rough cast teeth are comparatively easy to construct. 

It is possible, however, to construct a worm wheel in such a 
manner as to secure line contact, as in spur gearing. Referring 
to Fig. 164, it can be seen that when the single-threaded worm 
shown is rotated through 360 , any median section as A is moved 
forward an amount equal to the pitch of the worm wheel to a 
position B; and that rotation of the worm, in general, is equivalent 
to a translation of these sections backward or forward. The 
action is equivalent to translating a rack of similar proportions, 
and, in fact, if the worm itself is moved axially it will engage with 
the teeth of the worm wheel in the same manner as a rack does 
with a gear. In the involute system of gear teeth the rack has 
straight sides,* and this property is usually taken advantage of 
in making worm gearing, since a worm thread of such a cross- 
section is easily machined. The sides of the involute rack face 
are at right angles to the line of contact, a O b, Fig. 147, and hence 
the inclination of the sides to each other is 2 0, Fig. 147, and in 
the standard system 2 = 29°. If other planes such as M N be 
passed through the worm and worm wheel parallel to the median 
plane X X, Fig. 164, it will cut a trapezoid from the worm some- 
what different from that cut by the median plane. The rack- 
like action of these trapezoids would, however, be similar to those 
on the median plane, and it is clear that the shape of the worm- 
wheel tooth in the plane M N may be so made as to mesh cor- 
rectly with this new trapezoidal section. It is evident that if 
enough such sections be taken, a complete tooth outline may be 
formed that will give line contact with a worm across its full face. 
It is evident also that any other form cf worm thread may be 
similarly treated. 

The preceding discussion demonstrates the possibility of 
line contact in screw gearing, and suggests a method by which 



* See " Kinematics of Machinery/' by John H. Barr, page 125. 



TOOTHED GEARING 



397 



the teeth of such gearing could be drawn, and hence constructed. 
There is no practical value in actually making such drawings; but 
teeth having this property of line contact are automatically pro- 
duced by what is known as the hobbing process. A worm wheel 
of tool steel is made of the exact form of the desired worm. This 
worm is made into a cutter by cutting flutes across the face as in 
Fig. 1 68. This is known as a hob; and when hardened and 
tempered it is used as a milling cutter. The wheel blank, which 
has been turned to correspond to the outside of the teeth, is 
mounted in a gear cutter, or a special hobbing machine, and the 




Fig. 165. 



Fig. 166. 



hob is also mounted in correct relation to the wheel, but with the 
axes of the wheels a little greater distance apart than the required 
final distance. The hob is then rotated and at the same time fed 
toward the worm wheel till the proper distance between the axes 
is reached, thus cutting the teeth in the worm wheel in a very 
accurate manner. Sometimes the wheel is caused to rotate simply 
by the action of the hob, but much better results are obtained if 
it is driven positively, with the proper velocity ratio, from the 
cutter spindle by means of positive gearing. In heavy work the 
teeth of the wheel are roughed out or "gashed" before hobbing. 



398 MACHINE DESIGN 

Fig. 166 shows a worm wheel which has been hobbed, and its 
mating worm. Fig. 167* shows a form of wheel occasionally 
used where the wheel is sometimes rotated by hand or when the 
projecting teeth are undesirable. Such wheels may be hobbed, 
but are usually cut by the approximate method shown in Fig. 169, 
where a cutter is fed radially inward toward the axis of the worm 
wheel, producing what is known as a drop-cut wheel. In the 
Hindley worm the pitch line of the worm is curved to coincide 
with the pitch line of the wheel, thus obtaining contact on several 
teeth at the same time.t 

183. Velocity Ratio of Worm Gearing. The axial advance 
per turn of the worm thread is called the lead. Thus in Fig. 164 
the lead of the single-threaded worm shown is the distance, parallel 
to the axis, from any point on the tooth section A, to a correspond- 
ing point on the section B, and is equal to the circumferential 
pitch of the worm wheel. If the worm were double-threaded the 
lead would be twice this amount, or equal to the distance between 
corresponding points on A and C, and would then be twice the pitch 
of the worm wheel. The lead of the triple-threaded worm would 
be three times the pitch, and so on. If a single-threaded worm 
makes one revolution, a tooth of the worm wheel is moved a 
distance equal to the pitch. In the case of a double-threaded 
worm the tooth would be moved twice the pitch; and in general 
if N be the number of teeth in the worm wheel, and n the number 

... angular velocity of worm N 

of threads on the worm, then, : : — ; : — - = — . 

angular vel. 01 worm-wheel n 

Evidently a very great velocity ratio is possible with a compara- 
tively small worm-wheel. It is to be especially noted that the 
angular velocity ratio is independent of the diameter of the worm. 
The pitch of the worm wheel, which must be decided upon by 
consideration of the strength of the teeth, fixes the radius of the 
worm wheel for a given number of teeth; but the radius of the 
worm may then be varied to suit other conditions. 

* Figs. 165, 166, and 167 are reproduced from Browne & Sharpe's "Treatise 
on Gearing." 

t See "Worm and Spiral Gearing," by F. A. Halsey. 



TOOTHED GEARING 



399 



184. Efficiency of Worm Gearing. The general expressions 
for the efficiency of screws, deduced in Art. 54 of Chap. VII, 
apply also to worm-gearing. Since the worm thread is, usually, 
a so-called angular thread, equation 13 (a) of that Article would 
strictly apply. However, the inclination of the face of worm 
threads is so small that the error introduced in using the simpler 
equations (9) and (10) of that article, which were deduced from 
the square thread, is small. These equations show that the 
efficiency of all screw gears is a function of the angle which the 
thread makes with a plane perpendicular to the axis, and of 





*»--- 




Fig. 167. 



Fig. 168. 



Fig. 169. 



the coefficient of friction, assuming that the coefficient of fric- 
tion at the thrust collar is the same as at the tooth. 

One of the most valuable contributions to this subject is the 
experimental work of Mr. Wilfred Lewis.* The full lines in 
Fig. 170 have been plotted from the diagram on which he has 
summarized his results. They show clearly the increase of 
efficiency with increase of thread angle at all velocities. They 
also show a remarkable agreement with the theoretical equations 
of Art. 54. The dotted curve is reproduced from curve (2) of 
Fig. 52, and its close agreement with Mr. Lewis' curves is to be 
noted. This dotted curve was plotted for a value of jn = 0.05. 



* Trans. A. S. M. E., Vol. VII, page 297. 



400 



MACHINE DESIGN 



Mr. Lewis' calculated average value of this coefficient for a 
velocity of 20 feet per minute is 0.059 and for 10 feet per minute 
0.074. Curves (4) and (5) in Fig. 52 may, therefore, be taken 
as supplementary to those in Fig. 170, and may be used, as they 
were intended, for designing slow-moving and poorly lubricated 
screws. A theoretical curve plotted from equation (9) , Art. 54, 
with a value of /jl = 0.014 (which would be obtained only at 
high speeds), will coincide very closely with curve 1, Fig. 170. 



aoo 



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10 15 20 25 30 

Lead Angle in Degrees 

Fig. 170. 



35 



.40 



45 



This coincidence is closer than might be expected from the 
nature of the problem and the assumptions on which equation 
(9) is based. Mr. Lewis' value of // for these velocities * (200 
feet per minute) ranged from 0.026 to 0.015, his average value 
being 0.02. 

Mr.Halsey | has examined the design of a number of success^ 

* Velocity here means velocity of rubbing at the point of contact between worm 
and worm wheel. 

t See "Worm and Spiral Gearing," page 38. 



TOOTHED GEARING 401 

ful and unsuccessful worms used for transmitting power and 
found that every worm among those examined whose lead angle 
was greater than 1 2 — 30' was successful, and every worm whose 
lead angle was less than 9 was unsuccessful, and quotes Mr. James 
Christie, who has had considerable experience with this form of 
gearing, as giving 17 — 15' as the lower limit for successful design, 
which still further corroborates the general theory given. It is 
to be noted, on the other hand, that there is little to be gained in 
using a pitch angle above 30 , the increase in efficiency being 
very small, while the side thrust on the wheel is increased. It 
is not to be understood that it is never proper to design a worm 
with a lead angle less than 9 ; for there are many cases, not 
primarily for power transmission, and where the velocity is low, 
in which worms of less pitch are not only effective but neces- 
sary. In Mr. Lewis' experiments the worms ran in a bath of 
oil, and the efficiencies given include journal friction, the thrust 
being taken at the end of the worm shaft by a loose brass 
washer running between two hardened and ground steel washers 
(see Art. 104). 

The effect of the velocity of rubbing on the coefficient of 
friction of imperfectly lubricated surfaces, was noted in Art. 32, 
and Fig. 17 of that article indicates, in a general way, what may 
be expected with sliding surfaces: all experimental results going to 
show that the lowest coefficient was obtained at about 200 feet per 
minute. Mr. Lewis, as the result of his work, fixes 200 feet per 
minute as the point of maximum efficiency of worm gearing, 
which is in perfect accord with the general theory of lubrication. 
The surfaces of worm gearing, although running in an oil bath, 
must, from the nature of the contact, be classified as imperfectly 
lubricated surfaces. An increase of velocity may, up to a certain 
limit, decrease the coefficient of friction, but it is not possible at 
any speed, with the small amount of surface contact obtainable 
in screw gearing, to create a true oil film so that the load would 
be fluid-borne (Art. 33). 

185. Limiting Pressures and Velocities in Worm Gearing. It 
was stated in the last two articles that the best results are obtained 
from worm gearing when the rubbing velocity is about 200 feet 
26 



402 MACHINE DESIGN 

per minute and the lead angle not less than 12 — 30'. It is not 
always possible, however, to keep the design within these limits. 
Thus in order to obtain mechanical advantage (see Art. 64), it may 
be necessary to use a worm with a very small lead angle, and 
kinematic requirements may necessitate a much higher velocity 
than 200 feet at the pitch line. 

The allowable axial load that may be applied to a worm under 
varying velocities has not been very accurately determined, the 
law undoubtedly being complex (see Art. 32). Enough experi- 
mental work has been done, however, to show that the pressure 
varies, approximately, inversely with the velocity; or the law 
may be roughly expressed as W V = K, where W = the axial load 
on the worm, V = the velocity of rubbing in feet per minute, and 
K = a constant to be determined by experiment (see also Art. 
98) . In Lewis' experiments, made on cast-iron worms and worm 
wheels, running in an oil bath, it was found that the limiting value 
of K, i.e., where cutting began, was about 1,500,000. Smith and 
Marx* quote corresponding pressures and velocities, attributed 
to Stribeck, obtained with hardened steel worm and bronze 
worm wheel running in an oil bath, which give an average allow- 
able value of 690,000 for K. Bach and Roser ; experimenting with 
soft-steel worms and bronze worm wheels, succeeded in carrying 
a pressure of 800 pounds at a velocity of 1,700 feet per minute, 
which gives K = 1,360,000. It would appear, therefore, that for 
average conditions and bath lubrication of the worm it will be 
safe, for velocities up to 1,500 feet per minute, to take 

W V = 750,000 (14) 

The above discussion has reference to worms as ordinarily con- 
structed with straight-sided threads. Mr. Robert Bruce f has 
shown that if the sides of the worm are made concave sl much 
greater load may be carried. With improved threads of this form 
he has succeeded in carrying 25 tons at a velocity of 120 feet per 
minute, corresponding to K = 6,720,000. This great gain is due, 

* " Machine Design," page 301. 

t Proceedings of Institution of Mechanical Engineers (British), page 57 of 
tKeyear 1906. 



TOOTHED GEARING 403 

without doubt, to the improved lubrication obtained by what 
practically amounts to surface contact, between the mating convex 
and concave surfaces of the teeth. 

186. Design of Worm Gearing. In general, the strength of 
the worm exceeds the strength of the teeth in the worm-wheel; 
and where the worm is made of a harder material, which is the 
usual case, the wear is greatest on the worm-wheel teeth. It 
is usually sufficient, therefore, to design the wheel teeth alone, 
considering them as simple spur gear teeth as in Art. 173. In 
the case of rough-cast, or drop-cut teeth, it must be assumed 
that the entire load is carried by a single tooth; but in hobbed 
gearing it is safe to assume that the load is distributed between 
two, or even three, teeth, depending on the number of teeth in the 
wheel. 

Example. Design a worm gear to connect two shafts which 
are n inches apart, and to transmit 7^ H.-P. The velocity ratio 
is to be 20 to 1, the worm shaft is to make 320 R.P.M., the lead 
angle is not to be less than 15 , and the worm wheel is to be cut 
with a hob. 

The solution of problems in worm gearing must, generally, 
be tentative. If the velocity ratio is to be 20 to 1, the worm-wheel 
will have 20, 40, or 60 teeth, depending on whether the worm is 
single-, double-, or triple-threaded. It is difficult to obtain a high 
lead angle with a single-threaded worm without making a very 
large thread, therefore a trial assumption will be made with a 
triple-threaded worm, and 60 teeth in the wheel. Twenty inches 
may be taken as a trial diameter for the wheel, and the trial pitch 
circumference will therefore be 63 inches approximately. If the 
circumferential pitch be taken as one inch, the lead of the worm 
thread will be three inches, and can therefore be easily cut in a 
lathe. The corrected circumference of the wheel will then be 
60", corresponding to a pitch diameter of 19.11". The pitch 
diameter of the worm, with the given distance between centres, 

will be 2.9/; hence the tangent of the lead angle = 



* X 2.9 
0.33, or the lead angle is 18 — 15^ which is an efficient angle. 



404 MACHINE DESIGN 

The number of revolutions per minute of the worm wheel will 

320 
be =16. Hence the velocity of the worm wheel at the pitch 

T . 60 X 16 . . • 

line = = 80 feet per minute. The total axial thrust 

12 

on the worm will be — = 3,100 pounds. The velo- 

00 

city of rubbing equals the length of one turn of the worm 
thread multiplied by the number of revolutions per minute, or 

__ 7r x 2.9X320 7r x 2.0 X 320 

V = -, —. jT\ = =2SS ft. per minute. 

(cosi8' — 15") X 12 0.95X.12 

The product of velocity and axial pressure on the worm = 255 X 
3,100 = 790,000 which by equation (14) is a safe value, although 
somewhat high. 

The load may be considered as distributed between two teeth, 
and each tooth will have a face or length at the root at least equal 
to the pitch of the worm (see Fig. 164), or say 2.75". Hence 

the load per inch of face of tooth = — = <6o pounds. 

^ 2x2.75 

From the diagram, Fig. 158, it is seen that this load corresponds 
to a fibre stress of about 5,000 pounds per square inch with 1 
inch circular pitch. From equation (10) of Art. 175, however, 
it is seen that for the velocity, 80 feet, the allowable stress is 7,000 
pounds, hence the tooth has an excess of strength to provide 
against the wear, which falls heaviest on the worm wheel. 

From curve (1), Fig. 170, it is found that the efficiency is about 
90 per cent; hence the horse-power which must be supplied to 

7 • S 

furnish j.< H.-P. at the worm-wheel shaft will be = 8.4H.-P. 

' J 0.90 

= 277,200 foot-pounds per minute, or 866 foot-pounds per revolu- 
tion of the worm. The torque T, which must be applied to the 

worm-wheel shaft, will be T — = 1,650 inch-pounds. 

2 TZ 

The depth below the pitch line of a standard tooth of one inch 
circular pitch is, from Table XXV, 0.3857 inches; therefore the 



TOOTHED GEARING 405 

diameter of the worm at the root of the thread = 2.9— (2 X 0.3857) 
= 2.13", and from equation E, page 94, the torsional stress p s = 

16T 16 X 1,650 o , . , ... , . 

— ■= = - rf = 850 pounds per square inch, which is very 

low. The design may, therefore, be considered satisfactory if 
the worm is to be cut integral with the shaft. If, however, it is 
to be bored out and fitted over the shaft, further calculation as to 
the strength of the shaft which may be fitted is necessary. 

187. Thrust Bearings for Worms. An important frictional 
loss in worm gearing occurs in the thrust bearing, which therefore 
deserves special attention. The general discussion in Art. 104 
applies in this case. The type of bearing shown in Fig. 88 is 
much used, and of late ball bearings have met with considerable 
success in such places. 



CHAPTER XV 
FLYWHEELS AND PULLEYS 

1 88. Capacity of Flywheels. — There are two distinct types of 
flywheels; namely, those whose sole function is to absorb and 
redistribute energy, as noted in Articles 2, 4, and 6.2, and those 
which also act as a pulley or band wheel and transmit power 
continuously. When a flywheel is attached to a train of mechan- 
ism in which the supply of energy varies it tends to absorb any 
excess energy, thus having its velocity increased. When the 
work to be done is in excess of the energy supply, the wheel tends 
to furnish the deficiency at the expense of its kinetic energy, 
with a resulting reduction of velocity. Flywheels, therefore, to be 
effective must vary in velocity; the allowable amount of variation 
depending on the conditions of the case. Thus in engines 
driving electric generators, the variation from normal speed 
may be limited to one-half of one per cent, or less, while in such 
machines as punching machines, the variation may be as great 
as twenty per cent. 

If W be the weight in pounds of a body moving with a velocity 

of v feet per second, then the kinetic energy in foot-pounds which 

Wv 2 
the body possesses is K = where g = 32.2. If the velocity 

of the body be changed from v 1 to v 2 , the change in kinetic energy 
is the work which the body will do, or the energy it will absorb, 
depending on whether its velocity is decreased or increased. If, 
then, the work to be done or the energy to be absorbed with a 
given change in velocity is known, the necessary weight of the 
body may be found; for if K L be the kinetic energy of the body 
when moving with a velocity v lt and K 2 be the kinetic energy at a 
velocity v 2 , then the energy delivered or absorbed during a change 
of velocity is 

406 



FLYWHEELS AND PULLEYS 



407 



W v 2 W v.* W 

2g 2g 2g ~' W 

If the body is rotating around a fixed axis, the velocities of differ- 
ent points in the body vary as the distance of these points from the 

W 

axis. For this case the kinetic energy of the body is — p 2 co 2 

2 g 
where p is the radius of gyration, and w the angular velocity. 
Hence for rotating bodies equation (1) may be written 

W p 2 
E = K l -K 2 = -^ K 2 - <, 2 2 ) ... (2) 

2 6 

W p 2 
or since = /, the moment of inertia * of the body, 

o 

E = K t - K 2 = I (<** -»,«). . . . (3) 

In all cases of flywheel design the effect of the hub may be 
neglected, and in nearly all cases the effect of the arms is so small 
as to be negligible, and the rim only need be considered. When 
such is the case it is sufficiently accurate to take the mean radius 
of the rim R as the radius of gyration, and equation (2) becomes 
identical with equation (1) since, in general, R & = v. In the case 
of wheels with many heavy arms, or heavy disc wheels, and where 
it is desirable to compute the inertia effect of the wheel closely, 
as in direct driving of electric generators, equation (2) or equation 
(3) is applicable. In the case of a wheel with arms whose sides 
are parallel, or nearly so, it is to be noted that the square of the 
radius of gyration of the arms or p 2 is very nearly equal to y^ R 2 . 
Hence for this case, if W be the weight of the rim and W a the 
total weight of the arms 

E = K i -K 2 = ^-(W+ XW) K 2 - « 2 2 ) . (4) 

2 6 

Example (1). A punching machine is to make 30 strokes 
per minute and is to punch holes ffl in diameter in steel plate y 2 " 

* The student should distinguish clearly between the moment of inertia of 9 

solid body and the moment of inertia of an area. See Church's "Mechanics," Art 

W R 2 
86, page 91. In the case of a circular disc / = ■. 



408 



MACHINE DESIGN 



thick. Since the machine may be used for shearing also, it should 
be capable of punching a hole, or of doing the equivalent amount 
of work in shearing, at every stroke of the punch, continuously. 
The belt speed is to be about 600 feet per minute, and, from 
existing machines of the same type, it is known that the efficiency 
will not exceed 85 per cent. It is required to find the cross- 
section of the flywheel rim. 

Let Fig. 171 represent the machine under discussion. The 
mechanism in the head, A, is a slotted crosshead;* so that the 
punch M moves with harmonic motion. Let the diagram, Fig. 
171 (a), represent the path of the pin F, in the crosshead. When 




Fig. 171 (a). 



Fig. 171 



the pin is at b the punch enters the plate, and emerges from the 
lower face of the plate when the pin is at c. When the pin is at d 
the punch is at its lowest position, and has entered the die %"', at e 
the punch is withdrawn from the plate and at / is at its highest 
position. The pin, therefore, moves through an angle of 30 
while the work of punching is being performed. 

The preliminary layout also shows that the diameter of the 
driving pulley N should not exceed 18", and the mean diameter 
of the flywheel rim should not exceed 42". A preliminary 
estimate also fixes the ratio of the diameter of the pinion B to 
the diameter of the gear C as 1 to 6; hence the driving shaft will 
make 30 X 6 = 180 R.P.M. The circumference of the driving 

* See "Kinematics of Machinery," by John H. Barr, page 184. 



FLYWHEELS AND PULLEYS 409 

„ belt speed 600 
pulley = = -z- = 3.34 feet and, therefore, the diam- 
eter of the driving pulley will be 13" which is well within the 
limit set. The machine makes an energy cycle (see Art. 4) 
every stroke of the punch, or every six revolutions of the driving 
shaft. While the hole is being punched the flywheel is giving up 
energy to assist the belt, and during the remainder of the cycle 
the belt withdraws the punch from the sheet, and restores the 
wheel to normal speed. 

The greatest pressure which the punch must exert on the plate 
will be at the beginning of the punching operation, and will be 
equal to the area of the metal in shear multiplied by the shearing 

resistance; orP = 7rX~X — X 60,000* = 70,800 lbs. If the 

42 

belt had to exert this effort unaided, it would have to be of double 
leather 15 inches wide; hence the need of a flywheel. As the 
punch, passes through the plate the shearing resistance decreases, 
until it becomes zero as the punch passes out. The average 
pressure may therefore be taken as half the maximum, and the 

t 1 r 1 • , • • 70,800 1 1 
total work performed in punching is X - X — = 1480 

ft.-lbs. The work of withdrawing the punch from the sheet is 
small and may be considered as part of the frictional loss. Since 
the efficiency of the machine is 85 per cent the belt must supply 

T /|P\0 

' = 1740 ft.-lbs. every energy cycle. The energy delivered 
0.85 

by the belt per cycle is the product of the difference of the belt 
tensions (7\ — T 2 ) multiplied by the distance through which the 
belt moves (see Art. 131). Since the speed of the belt is 600 feet 
per minute, and the time of the cycle is -^ of a minute, the belt 
moves a distance = ■ s ^- = 20 feet per cycle. Hence (T t — T 2 ) 20 
= 1740 or T 1 — T 2 = 87 lbs. The effect of centrifugal 
force may be neglected at this belt speed, hence equation (8), 
Art. 131, gives/ = o.6t for an arc of contact of 180 and fi = 

* The shearing strength of a plate in punching is about equal to its tensile 
strength. 



4IO . MACHINE DESIGN 

0.3, where/ = the effective pull per inch of width of belt and t = 
tension per inch of width of belt on the tight side. In this class 
of machinery where excessive slipping of the belt is sure to occur, 
/ should be taken at not more than 40 lbs. per inch of width; 
whence 7=0.6X40 = 24 lbs., and the width of the belt = 

87 ,. -', 

— = 3>2 inches. 
24 6/ 

Since the pin, Fig. 171 (a), moves through 30 , or T V of a 
revolution, during the operation of punching, and since the 
punch makes 30 strokes per minute, the time consumed during 

the operation of punching will be — X — = —7- of a minute. 

12 30 360 

The belt, therefore, moves 600 X 3-g-^ = 1.7 ft. during the 

operation and supplies 87 X 1.7 = 150 ft.-lbs., leaving 1750 — 

150 ==■ 1600 ft.-lbs. to be supplied by the flywheel. The driving 

shaft makes 180 R.P.M., or 3 revolutions per second and, 

therefore, since the mean radius of the wheel =21 inches, 

2X^X21X3 
v t = = 33 feet per second. 

The allowable variation in velocity may be taken as 10 per cent, 
hence v 2 = 33 X 0.90 = 30 feet per second. Therefore, neglect- 
ing the effect of the hub and arms, the weight of the rim is from 
equation (1) 

w 2 8 E 2 X 32.2 X 1,600 

W = -% - 2 = ^ 5 = 545 lbs. 

v i - W 33 - 3° 

One cubic inch of cast iron weighs 0.26 lbs., hence the number of 

545 

cubic inches in the rim will be = 2,100. The mean cir- 

0.26 

cumference of the rim = 71X42 = 132 inches; therefore the 

2,100 , . . . , . . 

cross-section = =16 square inches, or a section 3^8 inches 

wide by 5 inches deep may be selected. 

In the above example it is quite easy to compute the amount 
of energy to be redistributed, from the conditions of the problem. 
In the more general case this cannot be done so readily, and 
methods such as those outlined in Articles 5 and 6 of Chapter 



FLYWHEELS AND PULLEYS " 411 

II must be employed, where the diagram representing work to be 
done is superimposed upon that representing the energy supplied, 
in such a manner that the excess and deficiency may be measured. 
Care should also be exercised that the solution covers a complete 
energy cycle in order that the solution may be based on the greatest 
excess or deficiency. Thus in Fig. 2 (d) the flywheel redistributes 
energy on both strokes, but the maximum excess of effort is 
represented by K and not by K x . Again in such machines as 
internal-combustion engines with " hit-and-miss " governors, 
giving a very variable energy supply, the design of the flywheel 
may have to be based on a hypothetical performance of the 
engine covering a number of successive strokes, or in other cases 
may be based on empirical constants which are the result of ex- 
perience. 

In some machinery, such as steam engines, it is desirable 
to limit the variation in velocity to a definite amount both above 
and below the mean velocity. If v be the mean velocity, v x and 
v 2 the maximum and minimum velocities respectively, then it is 
sufficiently accurate for most work to take v = % (y x + v 2 )> 
but the true relation between these quantities depends on the 
manner in which the velocity changes. For very exact work, as 
in parallel operation of alternating generators, it may be necessary 
to take this into account.* 

Example (2). Find the weight of a cast-iron flywheel neces- 
sary to limit the speed of the engine discussed in Art. 5 to a total 
variation of not more than 0.01, i.e., 0.005 above or below the 
mean speed. The wheel is also to act as a belt wheel, and the 
belt speed is to be about 4,000 feet per minute. The sides of 
the arms are to be parallel and their effect is to be considered. 

Since the engine makes 160 R.P.M., the circumference of 

the wheel = -~ — = 2K feet. Therefore the diameter of the 

160 J 

wheel = 8 feet, which may be taken without great error as a 
mean diameter of the wheel rim, since the thickness of the rim 
will not be great. The face of the wheel should be at least 16 

* See a paper by I. J. Astrom, Trans. A. S. M. E., Vol. XXII, page 972. 



412 MACHINE DESIGN 

inches wide to give the necessary width of belt. A preliminary 

layout gives 6 arms, each weighing 170 pounds, or W & = 170 X 6 = 

160 
1,020. Let n = the revolutions per second = •—— and R = the 

60 

radius = 4 feet; from Art. 5, E — 4,620 foot-pounds. Then «> 

= 27tn — 27tX -7— — 16.8 radians. 
60 

id _|- a> 

Now, oj = and hence from the conditions of the 

2 

problem, n> x = 1.005^ = 1.005 X 16.8 = 16.88 ; and oj 2 = 
0.995 w = 16.7. Therefore 

( Wl 2 - «, 2 2 ) = (16. 88 2 - 16. 7 2 ) = 6.1 

2 gE 
From Equation (4), (W + % Wj = R2 , 2 _ a) 2 ) 

2 gE W„ 2X32.2X4,620 1,020 

or W = „ 2 , 2 27 = > = 2,720. 

R 2 K 2 ~ ^2) 3 16x6.1 3 M 

Therefore W + W a — 2,720 + 1,020 = 3,740 lbs., to which must 

be added the weight of the hub to obtain the total weight of the 

wheel. Since the rim is to be 16 inches wide and the mean 

2,720 
diameter is to be 96 inches the thickness will be 



o. 26 X n X 96 X 16 
= 2.2 inches or say 2% inches. 

189. Practical Coefficients. It is evident that the ratio of 
the energy to be absorbed, to the total energy supplied per 
energy cycle, will vary in different machines, and also in the 
same machine under different conditions. Thus in the punch- 
ing machine the flywheel absorbs and redistributes nearly the 
entire energy supply per cycle, while in the steam-engine example 
the amount absorbed is about one-third the total. It is readily 
seen that in the steam engine this ratio will vary with the point 
of cut-off, steam pressure, weight of reciprocating parts, etc., and 
therefore, in general, tabulated values of this ratio are deceptive 
unless they refer to specific conditions. It is to be noted that 
the weight of the flywheel is directly proportional to the energy to 
be absorbed and inversely proportional to (v* — v 2 2 ). The latter 



FLYWHEELS AND PULLEYS 413 

is usually a small quantity and, therefore, if E is large the weight 
of the flywheel may be excessive, which is undesirable because 
of the cost, and also because heavy wheels bring great loads on 
the bearings, causing frictional losses. For this reason it is 
always desirable so to arrange the sequence of events in the energy 
supply and work to be done as to minimize the excess energy to 
be absorbed. This is illustrated in Art. 6, Fig. 5, where the area 
K may be greatly decreased (or increased) by changing the relative 
positions of the crank pins. This procedure is of great import- 
ance to avoid wheels of great weight in large steam engines when 
variation in velocity must be closely restricted. 

The allowable variation in velocity is fixed with reference 
to the character of the work to be done. It is evident that some 
classes of work require much more constant velocity than others, 
and experience has shown what the limits in variation of 
velocity may be for successful operation. The following limiting 

values of the proportionate variation represent average 

practice. The particular case of direct driving of alternating 
generators in parallel must, in general, be treated with reference 
to the allowable variation per pole, and when, therefore, the 
number of poles is great the total allowable variation is corre- 
spondingly small. 

TABLE XXIX 

Values of 

v 

For punching machines and similar machines . . o . 10 to o. 15 

For engines driving stamps, crushers, etc o . 20 

For engines driving pumps, saw mills o .03 to o .05 

For engines driving machine tools, weaving and paper 

mills o .025 to o .03 

For engines driving spinning mills for coarse thread .... o .016 to o .025 

For engines driving spinning mills for fine thread 0.01 to 0.02 

For engines driving single dynamos o .007 

For engines driving alternators in parallel o .003 to o .0003 

190. Stresses in Flywheels. The velocity of the rims of all 
flywheels is, from the nature of their requirements, very high. 
If the wheel is to act as a band wheel, the desirability of obtaining 



414 



MACHINE DESIGN 



high belt speed (Art. 134) brings the peripheral velocity up to 
4,000 or 5,000 feet per minute. It has been shown that the 
capacity of a given wheel is proportional to the square of its 
velocity and, therefore, when the wheel is to act as a flywheel 
alone, economy in the use of material, or the limiting of the 
external dimensions, makes high speed very desirable. Great 
care should be used in the design of such wheels, for a flywheel 
which breaks at normal speed is exceedingly dangerous to life and 
limb, and when such wheels "explode" or break from overspeed- 
ing, the results are usually very disastrous. 

Unfortunately, mathematical analysis of the stresses in fly- 
wheels and pulleys is not satisfactory or conclusive. In the case 
of small wheels cast in one piece, unknown shrinkage stresses 
of great magnitude may exist, which render useless any refined 
calculations. In large wheels built up of sections, the presence 
of joints vitiates any calculations based on the elastic theory of 
the strength of materials; and when the parts are of cast materials 
and of large sectional area, there is no assurance that the 
character of the material is uniform throughout. It is impor- 
tant, however, to understand fully the general character of the 
stresses even when no accurate computations can be made as to 
their magnitude. 

Consider that the rim of the wheel in Fig. 172 is free to expand 
radially, the arms exerting no restraining force in a radial direc- 
tion. If the wheel be rotated on its axis the action of centrifugal 
force is such as to cause an outward pressure on every part of the 
rim, in exactly the same manner as in a boiler shell acted on by 
an internal pressure (see Art. 78) ; the rim expanding until the 
tensile stress induced in any section A A, Fig. 172, balances the 
tendency of the wheel to separate along that section. If, on the 
other hand, the arms are rigidly attached to both hub and rim, 
and are so inelastic that their stretch, under the action of the 
centrifugal pull due to their own mass and that of the rim, is 
negligible, it is clear that they may be placed so close together 
that the rim cannot exoand, and practically no stress will exist 
in the rim, the centrifugal action being balanced by the stress in 
the arms. 



FLYWHEELS AND PULLEYS 



415 



Flywheels approximating both of these conditions are some- 
times built, but in the most usual case the arms stretch a certain 
amount and are not placed close together, so that a condition re- 
sults similar to that shown, in an exaggerated manner, in Fig. 
173. Here, the arms, though stretching somewhat, do not 
stretch enough to allow the rim to expand freely, and, therefore, 
the hoop tension is somewhat less than that in the free ring. The 
section of rim between each pair of arms is so long that it becomes 
a beam fixed at the ends and loaded uniformly by the unbalanced 
centrifugal action; the greatest bending moment being at the arm, 
and abending moment of half the maximum occurring at the centre 




Fig. 172. 



Fig. 173. 



of the span. The maximum tensile stress will be the sum of the 
hoop tension and the tensile stress due to the bending action. 
The relative values of the hoop tension and bending stress will, 
evidently, depend upon the amount which the arms stretch. 
If they should stretch enough owing to their own centrifugal force, 
so that the rim expands freely, no bending action will occur; 
while if they are so inelastic as completely to restrain the rim, 
no hoop tension will be induced, but the full centrifugal force 
will be applied to bend the rim. With any intermediate amount 
of stretch of the arms the rim will be held in equilibrium, partly 
by the hoop tension and partly by the restraining action of the 



41 6 MACHINE DESIGN 

arms, the latter being a measure of the unbalanced centrifugal 
force of the rim, and of the bending stress caused thereby. Since 
the expansion of the rim is directly proportional to the stretch of 
the arms it is clear that the hoop tension is also directly propor- 
tional to the stretch. If, for instance, the arms stretch one 
quarter the amount necessary for free expansion, the hoop tension 
will be one-quarter that due to free expansion, and the bending 
stress will be proportional to three-quarters of the centrifugal 
force of the rim. The mathematical relation which exists between 
these stresses is complex, and will of course vary with the relative 
size and shape of the rim and the arms. If the rim is of a wide 
thin section, and the arms are few, the bending stress may be very 
serious. Professor Lanza* has shown that, with the proportions 
ordinarily used, the arm, theoretically, stretches about three- 
quarters the amount necessary for free expansion. It is also to 
be noted that if the wheel is to act as a band wheel, and has a 
wide thin rim, the bending action on the arms as at B, Fig. 173, 
still further distorts the rim and increases the bending on the 
forward side. 

Let D = the mean diameter of the rim in feet. 
Let R = the mean radius of the rim in feet. 
Let t = the thickness of the rim in inches. 
Let v = the velocity of the rim in feet per second. 
Let w = the weight of the material per cubic inch. 
Let / = the length of the rim between arms in inches. 
Consider a section of the rim one inch wide on the face. The 
centrifugal force per unit of length (1"), circumferentially, of 

IV t IT 

this section is c = — — • and, therefore, by Art. 78, the total load 

Kg 

wtv 2 
which tends to separate such a ring along a diameter is — — 

X 12Z), and the unit stress in the section, if no bending exists, 
is therefore, 

i2ivtv i D 12 wv 2 1? . , . . . .. 

p. = = = — , nearly, for iron wheels . (<) 

ri 2tRg g 10' J ' vo/ 

* Trans. A. S. M. E., Vol. XVI, page 208. 



FLYWHEELS AND PULLEYS 477 

The maximum bending moment in the rim occurs at the 

. cl 2 * 

arms, and its value is M = considering the rim as a 

12 ° 

straight beam. The stress due to this bending moment when no 
hoop tension exists is therefore, 

Me cPe 

where e is the distance to the outer fibre, and 7| the moment of 
inertia of the cross-section of the elementary ring. 

If now the stretch of the arms be taken as three-quarters 
that necessary for free expansion of the rim, the total unit tension in 
the rim will be 

( ?>v 2 cPe\ 

P-XP l + Xt*-{ I £ + #j) • • • (7) 

12 7T D 

if n be the number of arms in the wheel, / = , and if the 

n 

cross-section of the rim be rectangular j = -5 whence equation 

(7) reduces to 

40 tn / Hn 



*-fc + VJ = 3^1-; + 0.025) . . (8) 



For t = — , v = 88, D = 4 ft., and n = 6. Professor Lanza 
10 

finds the stress due to hoop tension = 575 and the stress due to 
bending = 5,060 or p the total stress = 5,635. For the same 
data equation (8) gives HPi — 581 and %p 2 = 4,600, or a total 
stress p = 5,811, which agrees quite closely. 

The above equation may be used for checking, roughly, the 
allowable stress in flywheel rims, but implicit faith must not be 
placed upon it for the reasons given in the first paragraph of this 
article, and all results obtained from this or similar formulae 
should be checked by successful practice wherever a doubt arises. 
The equation does, however, show clearly that in wheels having 

* See Table I, case 17. 

fit should be noted that this I is for a unit (1") of width of rim and not for 
the entire cross-section. 
27 



418 MACHINE DESIGN 

thin rims, or few arms, the bending stress is much greater than 
that due to hoop tension, and care should be exercised ac- 
cordingly when such wheels must run at high speed. Equation 
(5) is often taken as a basis for the design of flywheels, using, 
therewith, a large factor of safety to cover uncertainties. If p x 
in equation (5) be taken as 1,000 (a factor of safety = 20), then 
v = 6,000 feet per minute, and this is found to be a safe peripheral 
speed for ordinary cast iron wheels. It is to be noted, however, 
that this speed is safe only because experience has shown it to be 
so, and not, as will be seen, because the stress is necessarily as 
low as 1,000 pounds. 

Example. Compute the stress in the rim of the cast-iron 
flywheel discussed in example (2) of Art. 188, assuming that the 
arms stretch three-quarters the amount necessary for free expansion 

4,000 
of the rim. Here n = 6, / = 2.2, D = 8 and v = — — = 66.6. 

00 

D 



.'. from (8), p = 3 v 2 (— 2 + 0.025) = 

(8 \ 
— 2 + 0.025 J = 1,668 lbs. per sq. in. 



The stress, if based on equation (5), would be 444 pounds per 
square inch. 

When a flywheel is being accelerated from rest, or when the 
energy supply is suddenly cut off, as it may be in a steam engine, 
the arms may be called upon to carry the full torque load. Each 
arm of a wheel with a very stiff rim approximates a cantilever 
beam fixed at one end, free but guided at the other, and car- 
rying a concentrated load at the free end (see Table I, case 7). 
If the rim is thin and flexible, the arms approximate a simple 
cantilever loaded at the free end. In addition, the arm is sub- 
jected to a tensile stress due to the centrifugal action of its own 
weight, and that part of the rim which it supports, so that appar- 
ently equation M (Table VI) applies. The direct stress is 
difficult to compute, however, and since the bending stress in 
the simple cantilever is twice that of a cantilever with the free end 



FLYWHEELS AND PULLEYS 419 

guided, it is considered sufficiently accurate to compute the arm 
as a simple cantilever and neglect the direct stress. 

Let P = the greatest force due to the belt pull at the rim. 

Let a = the length of the arm. 

Let n = the number of arms. 
Then from J (Table VI) : 

Pae 

* = ^T <9) 

from which the stress p, or the moment of inertia /, may be 
determined. The stress allowed should not exceed 2,000 pounds 
per square inch, for cast iron, on account of the uncertainties 
of the case, and a lower value is sometimes desirable. The 
statement sometimes made that the arms should be as strong 
against bending stress as the shaft is against torsional stress, is 
misleading as, in general, shafts are designed for stiffness, and 
not for torsional strength. The shaft of a steam engine may 
have to be very large to avoid excessive deflection and, as a 
consequence, may have great excess of torsional strength. 

191. Construction of Wheels. Flywheels and band wheels, 
for velocities below 5,000 feet per minute, are usually made of 
cast iron on account of low cost. For higher velocities steel 
castings are used, and in extreme cases wheels made of steel 
plates, or wire- wound wheels have been constructed. Equation 

(5) may be written v = 1.64 -J ^0. The allowable unit tensile 

™ w 

strength divided by the weight per cubic unit is, therefore, a 
measure of the value of the material for this purpose. For this 
reason some woods are superior to cast iron for wheel rims, and 
cast-iron wheels which have burst have been successfully re- 
placed with wheels having rims made of wood.* 

Difficulties in transportation limit the diameter of wheels 
cast in one piece to about ten feet, and the diameter of wheels 
cast in two parts to about twenty feet. Wheels from about six- 
teen feet in diameter upward are usually made in several sections. 
Small flywheels and band wheels are usually cast in one piece, or 

* Trans. A. S. M. E., Vol. XIII, page 618. 



420 



MACHINE DESIGN 



made in two parts for convenience in erecting. - In either of the 
latter cases unknown shrinkage stresses will most probably exist. 
These shrinkage stresses are sometimes relieved by casting the 
hub in several pieces, each piece being cast integral with one or 
more arms. The openings between the parts are afterward filled 
with lead, and rings are shrunk upon the hub to hold the parts 
in place. Experience shows that solid cast-iron wheels, when 
properly proportioned, are safe up to 6,000 feet per minute which, 
fortunately, is also about the limit of efficient belt speed. If, how- 
ever, the wheel has a very wide thin rim it cannot be considered 
safe at this speed, particularly if balance weights are attached to 
the rim between the arms, thus increasing the centrifugal bending 




Fig. 174. 



Fig. 175. 



force. If joints exist in the rim, their relative strength must be 
considered. Band wheels of wrought-steel construction can now 
be obtained up to about 4 feet in diameter; they are light and 
strong, and are rapidly coming into favor. 

Where speeds above 6,000 feet per minute are necessary, 
wheels such as shown in Fig. 174 are sometimes built. Here 
the rim and hub are of cast iron, each cast in one piece, and the 
spokes are of steel. The spokes are placed in the mould, and 
the metal poured around them, so that on cooling they are gripped 
very firmly. The spokes are placed close together so that there 
is practically no bending of the rim, and the rim is also prevented 
from expanding freely. Wheels of this construction are used for 
large band saws at velocities above 10,000 feet per minute, 
under heavy service, with perfect success. 



FLYWHEELS AND PULLEYS 421 

In Fig. 175 the rim is cast separately in one or more pieces. 
The arms do not constrain the rim radially, but leave it free to 
expand. The stresses in the rim when cast in several pieces so 
that shrinkage is not a factor are those due to centrifugal force 
only, and the arms are simple cantilevers. Wheels of this charac- 
ter have been used with success in rolling-mill work.* Figs. 174 
and 175 illustrate wheels which correspond closely to the limiting 
types discussed in Art. 190. The construction of most wheels 
lies between these types. Fig. 176 illustrates a band wheel with 
the arms and hub cast in one piece and the rim in sections. The 
joints in the rim are simple flange joints, placed midway between 
the arms. This is the most dangerous location possible, on 
account of the added bending effect due to the centrifugal force 




Fig. 176. Fig. 177. 

of the flanges which add to the mass without contributing to the 
strength. The best location is at the arm, and many wheels are 
built thus, the arm being bolted to each segment, and the seg- 
ments themselves bolted together as well. Where the joint is 
placed between the arms, it should be about one-quarter the 
length of the span away from the arm, as at A, Fig. 176, where, 
by the theory of elasticity, the bending moment is zero. Fig. 
177 shows a heavy flywheel with an arm and a segment of the 
rim cast together. The arms are secured in the hub by means 
of fitted bolts. The hub may be solid or the flange on one side 
may be movable axially so as more firmly to clamp the 
arms. The segments are held together at the rim by means 
of links of rectangular cross-section shrunk in place. This 
construction is very common. Occasionally links are also 

* Trans. A. S. M. E., Vol. XX, page 944- 



422 



MACHINE DESIGN 



shrunk into recesses on the outer face of the wheel. In Fig. 178 
the segments are held together by T-headed links, sometimes 
called " prisoners," shrunk in place. The segments are joined 
at the arms, which are fastened to them by through bolts. 
This construction is simple and the machining is easier than with 
flanged connections. The construction of the hub is similar to 
that in Fig. 177. 




Fig. 178. 



Fig. 179. 



It is evident that the manner of joining segments in built-up 
wheels is most important. Wheels seldom fail at the hub. 
Wheels with thin, wide sections are almost always joined by 
flanges as shown in Fig. 180. When such joints are used they 
should be well ribbed for stiffness, as indicated, and the bolts 
should be placed as near the rim as possible, so that the lever 
arm a shall be as great as possible compared to the arm b (see 



Hib' 



Fig. 180. 




Fig. 181. 



Fig. 182. 



Art. 63). A much better arrangement is shown in Fig. 181, 
where an arm is placed on each side of the joint. This is par- 
ticularly applicable to wheels cast in two parts. It may be noted 
that thin rims are often stiffened by light circumferential ribs at 
the outer edges. Mr. A. K. Mansfield has pointed out (Trans. 
A. S. M. E., Vol. XX) that these ribs maybe a source of weakness. 



FLYWHEELS AND PULLEYS 



423 



The greatest bending moment is near the arm where these ribs 
are on the tension side of the beam. A rim having such ribs is 
not necessarily as strong against bending in this direction, as one 
of rectangular cross-section having the same area ; and when 
ribs are used the section modulus should be calculated. 

The prisoner link shown in Fig. 178 has certain advantages over 
the link shown in Fig. 177. It is evident that the depth of the 
recesses in Fig. 177 is limited, while in Fig. 178 the slot can extend 
entirely across the section and the link can be made as wide as 
the rim itself. Furthermore, it is possible to machine both wheel 
and link in Fig. 178 accurately, which is difficult to do with the 
construction in Fig. 177. This permits of greater accuracy in 
computing the initial stress induced in the link by shrinking it 
in place, the importance of which has been noted in Art. 77. If 
the rim be made I-shaped,* as in Fig. 182, the links can be so pro- 
portioned that the joint will be as strong as the rim proper or 
even stronger. 

While, evidently, the relative strength of the joint com- 
pared to the solid rim will vary with the exact proportions selected, 
average practice gives the following apparent values: 

Flanged joint, bolted, midway between arms 25 

Flanged joint, bolted, at end of arms 50 

Linked joint as in Fig. 177 (Jo- 
Linked joint as in Fig. 178 65 

Linked joint as in Fig. 182 1 . 00 

Solid rim 1 . 00 

It must not be inferred from the above that a solid 
rim is necessarily the best; as, obviously, a wide thin rim 
with unknown shrinkage strain may not be as safe as a narrow 
deep rim of the same sectional area if held together by a good 
joint. 

For extreme velocities, wheels built up of steel plates, or 
wheels with rims made of plates fastened to a central spider made 
of steel castings, are now used. Fig. 179 shows a flywheel of the 
latter type used in rolling-mill work (see Power, Feb. 4, 1908). 
The rim is made of laminations held to the spider by dovetails, 

* See Trans. A. S. M. E., Vol. XX, page 944. 



424 MACHINE DESIGN 

as shown, the laminations being assembled with overlapping 
joints. Heavy outside plates clamp the whole structure together 
by means of through bolts. In the particular case noted above, 
the velocity of the wheel rim is 250 feet per second. Descriptions 
of a number of examples of such wheels are to be found in the 
Transactions of the A. S. M. E., the magazine Power, and other 
periodicals. Wheels for great speed have also been constructed 
by winding the rim with many turns of steel wire. 

The rotors of some forms of electric generators, steam turbine 
rotors, and similar revolving members are often loaded as shown 
at W, Fig. 176. Such loads add to the centrifugal force acting 
on the rim, but do not add to the strength of the rim. Due al- 
lowance should be made in such cases; particularly if the load 
or loads are placed near a joint as shown in Fig. 176. The 
teeth of gear wheels constitute such a load, and if the wheel is large, 
and the peripheral speed high, this should be considered. Balance 
weights, placed between the arms, should be carefully considered, 
especially when the rim is thin and the velocity high. 

192. Experiments on Flywheels. The best experimental data 
upon the strength of flywheels are from tests conducted by 
Professor Benjamin and reported to the A. S. M. E.* While 
these experiments were made to determine the bursting speed of 
small cast-iron wheels only, and throw no light on the increase of 
stress with an increase of speed,, they are very valuable as indicat- 
ing the manner in which various types of wheels fail. Being 
conducted on small wheels, due allowance must be made for the 
difference in quality between the metal of small and large castings 
in estimating probable bursting stresses. These experiments go 
to show that solid cast wheels will burst at a peripheral velocity 
somewhere near 400 feet per second, and such wheels are safe only 
at a velocity of not more than 100 feet per second. Rim joints 
midway between the arms, particularly the common flange joints, 
were found to reduce the strength materially. The strength 
of various joints was found to be about as tabulated in Art. 191. 

* See Trans., Vols. XX and XXIII. See also "Machine Design," by C. H. 
Benjamin. 



FLYWHEELS AND PULLEYS 



425 



Extra loads, such as balance weights located between the arms, 
were found to be very dangerous, on account of the added bending 
effect. 

193. Rotating Discs. If the radial depth of a wheel rim be 
great compared to its axial width, the equations deduced in the 
preceding articles do not apply, the difference being analogous 
to that existing between thick and thin cylinders. Mathematical 
analysis of the stresses in a rotating disc, in common with those 
existing in thick cylinders under internal pressure, are complicated 
and not altogether satisfactory. Experimental data, corroborat- 
ing the theories, are also lacking. A full mathematical treatment 
of these stresses is beyond the scope of this treatise, and only 
enough will be inserted to show the general character of the 
problem. 

When a disc of uniform thickness is rapidly rotated on its 
axis, the principal stresses induced are a tangential tension, and 
a radial stress, at every point in the disc. 

Let r 2 = the outer radius of the disc in inches. 

" r x = the inner radius of the disc in inches. 

" r = the radius at any point. 

" X = Poisson's ratio = )/$ for steel and % for cast iron. 

" N = revolutions per minute. 

" w = weight of one cubic inch of the material. 

" p '= the tangential stress at any radius r. 

" p 1 = the radial stress at any radius r. 
Then it can be shown* that for a flat disc of uniform thickness, 
having a hole at the centre of radius r v 

^ = o.ooooo 3 55wiV 2 [(3 + A)(f 2 H^ 2 + ; ^)-(i+3 ; > 2 ](ii) 

and p' = 0.00000355 wN 2 [(3 + ; )(^2 2 + ^ 2 - ! ^r--^)] (12) 

For a solid disc 

p = 0.00000355 w N 2 [ (3 + A) r 2 - (1 + 3 /) r 2 ] . (13) 
and p' = 0.00000355 wN 2 [ (3 + /) ir 2 - r 2 ) ] . (14) 



* See "Theory of the Steam Turbine," by A. Jude, pages 192 and 204. The 
notation and units have been changed to correspond with those used in this text. 



426 MACHINE DESIGN 

It is to be noted that the radial stress is less at any point than 
the corresponding tangential stress; and an examination of 
equation (n) shows that this tangential stress is a maximum at 
the surface of the bore and a minimum at the outer periphery. 
At the surface of the bore or where r = r , the stress 

p = 0.00000355 co N 2 [ (3 + /) (2 r 2 + r 2 ) - (1 + 3 X) r 2 ] 

If now r x be taken so small that r 2 is negligible, it appears 
that the tangential stress is 

p = 2 X 0.00000355 w N 2 [ (3 + X) r 2 2 ] 

which is just twice that obtained by making r = in equation 
(13). The effect of even a very small hole at the centre of a 
rotating disc is, therefore, to increase the stresses greatly. 

Example. A circular steel saw X inch in thickness and 80 
inches in diameter has a hole 4 inches in diameter in the centre 
and runs at the rate of 500 R.P.M. Determine the tangential 
stress at rim and also at the hole* 

Here AT" = 500, w = 0.28, ^ = %, r 2 = 40, and r x = 2. 
Whence in (11) making r = r 2 = 40 the tangential stress at the 
rim is 

p = 0.00000355 X 0.28 X 5oo 2 [ (3 + X) (40 2 + 2 2 + 2 2 ) - 
(1 + 1) 40 2 ] = 535 lbs. per sq. in. 

and at the hole making r = jr t = 2. 

p = 0.00000355 X o. 28 X 500 2 [ (3 + X) (4o 2 + 2 2 + 40 2 ) - 
(1 + 1) 2 2 ] = 2,643 lb s - per sq. in. 

The foregoing equations, (11) to (14), are deduced on the 
hypothesis that the material is perfectly elastic and homogeneous. 
It is clear that they cannot be intelligently applied to built-up 
wheels of the disc type, and must also be applied with caution to 
brittle materials. They are of great value, however, in showing 
the general character of these stresses and the location of the 
greatest stress, thus indicating the shape which discs should have 
for uniform strength; for a brief reflection will show that such 
discs must be thickened at the centre to reduce the stress at that 



FLYWHEELS AND PULLEYS 



427 



point. For complete mathematical analysis of discs of different 
shapes, reference may be made to the various works on the steam 
turbine. It is evident that great care should be used in selecting 
and working the material for high-speed discs. Rolled sheets 
are not good for very high speeds on account of their seamy 
structure, which is conducive to incipient cracks, and cast ma- 
terials of brittle structure must be of first-class quality. Discs 
forged down from much thicker ingots give the safest construction. 

References : 

"The' Theory of the Steam Turbine," by A. Jude. 

"Steam Turbines," by L. French. 

"The Steam Turbine," by Dr. A. Stodola. 



CHAPTER XVI 
MACHINE FRAMES AND ATTACHMENTS 

194. Stresses in Machine Frames. Since machine frames 
must, in general, receive the reactions from the forces applied 
to the various moving members by the energy transmitted, it is 
obvious that the stresses induced in frame members are, in most 
cases, very complex and beyond mathematical analysis. If it is 
essential that the moving members be held in accurate alignment, 
as in the case of machine tools, the predominating requirement 
for the frame is stiffness and not strength. For these reasons 
the design of machine frames, in general, must be governed largely 
by judgment and experience, the cases where complete mathe- 
matical analysis is possible being rare. However, even in cases 
where judgment must be the guide, it is not only helpful, but 
sometimes necessary to check, as closely as possible, the stresses 
in certain important sections, by applying those fundamental 
formulae of Table VI, page 94, which apparently fit the circum- 
stances. In all cases, what may be termed a " qualitative 
analysis" of the frame is very desirable as a guide in properly 
distributing the material, and in determining the forms of the 
various sections. 

If the character, value, and line of action of every force acting 
upon a given section are known, the stresses in the section can be 
determined by applying the fundamental requirements for static 
equilibrium of the section, namely: — 

(a) The algebraic sum of all horizontal component forces 

must = o. 

(b) The algebraic sum of all vertical component forces must 

= o. 

(c) The algebraic sum of all the moments must = o 

4.28 



MACHINE FRAMES AND ATTACHMENTS 



429 



The stress, in any direction, at any point, will be the algebraic 
sum of all the stresses acting in that direction, at that point, as 
found by applying (a), (b), and (c). It is impossible to make a 
classification of machine frames that would be of any service, but 
the principles will be illustrated by applying them to typical cases. 
It is to be noted that it is seldom possible to find the required 
dimensions of a section, directly, by solving the particular equa- 
tions from Table VI which apply; but, in general, the section 
must be assumed from the conditions given, and then checked 
for strength or stiffness. 

Fig. 183 illustrates a type of frame which is quite common 
and known as an open frame. It is one of the few types where 




Fig. 183. 



Fig. 184. 



a mathematical analysis can be made with some degree of com- 
pleteness. In the case of a punching-machine frame as illustrated 
in Fig. 183, great stiffness is not essential and the design may 
be based on the strength required. Suppose the frame to be 
outlined as shown so that the dimensions of the cross-section at 
any place may be assigned. Evidently, if the stresses are checked 
at the sections BC, DE, and FG, the strength of the frame will be 
fully determined. 

In the section BC, whose gravity axis is at O x , consider the 
portion of the frame above BC as a free body. It is in equilibrium 
under the action of the exterior force P, due to punching, and the 



430 MACHINE DESIGN 

internal forces exerted upon it by the lower half of the frame, 
There are no horizontal forces. The vertical force P must be 
balanced by an equal and opposite force at the section BC, which 
induces a tensile stress uniformly distributed over the section, 

P 

the intensity of which is p t = ~r pounds per square inch, where 

Jx 

A is the area of the section. The only moment acting on the part 

is Pa, due to the action of P, which tends to rotate the upper part 

of the frame around lf the gravity axis of the section, causing a 

resisting tension at B, and a resisting compression at C. The 

maximum intensity of these flexural stresses is given by the 

fundamental equation for flexure in beams (see /, Table VI), or 

Pae 
p 2 = —f — where e is the distance from 1 to the outer fibre 

and 7 X is the moment of inertia of the cross-section around the 
axis O x . The greatest tension will therefore be at B and its value 

P Pal 
will be p = p 1 + p 2 = — + -, which is equation M of 

Jx € 

Table VI. This is, therefore, a case of combined flexure and 
direct stress, which is fully discussed in Art. 19, Chapter III. 

Consider next the section DE, whose gravity axis is at 2 , 
and suppose the part of the frame at the left of DE to be a free 
body. There are no horizontal forces and the vertical force P 
must be balanced by a vertical tensile pull upon the upper part 
of the frame by the lower part. The resultant of this tensile 
pull, which is distributed uniformly over the whole section, may 
be represented by 2 K acting at the centre of gravity. This 
force may be resolved into the components HK = P x perpen- 
dicular to DE, and producing a tensile stress at right angles to 
the section and P 2 = 2 H parallel to DE and producing a 
shearing stress along the section. The only moment acting upon 
the section is that due, as before, to P, whose moment arm is a 2 . 
The tensile and compression stress due to this moment, as deter- 
mined by equation 7, Table VI, may be combined with the direct 
stress P 1} as in the section BC, to find the maximum tensile or 
compression stress. The shearing stress is 



MACHINE FRAMES AND ATTACHMENTS 431 

P 2 
p s = ~ where A 2 = the area of the section DE. 
A 2 

This is usually small and may be neglected except near the 
ends of the beam as in the section FG (see Art. 14, Chapter III). 

Consider last the section FG. As before, there are no hori- 
zontal forces, but the vertical force P must be balanced by a 
vertical resisting force which induces a shearing stress at the 

P 

section. The intensity of this shearing stress is — , where A a 

A 3 

is the area of the section. Since the area of the section is much 
smaller than at DE or BC, it is advisable to compute its value. 
The moment Pa 3 is balanced as before by the resisting moment 
of the section and the resulting stress may be computed by equa- 
tion /, of Table VI. Evidently these general principles may be 
applied to any section. 

Fig. 185 illustrates an open frame as applied to a power 
riveter. The rivet which is to be " driven' ' is placed between the 
dies D and D v and pressure is applied to the movable die D, by 
means of the power cylinder R. The pressure which is applied 
may be very great (150 tons or more), and unless the jaws are 
properly designed they may spring so much that the dies will 
fail to align properly, and faulty work will result (see Art. 53). 
Stiffness and not strength is, therefore, the essential factor in 
the design; for if the parts are stiff enough they will be, in 
general, strong enough. The yielding which most affects the 
ilignment is that due to the bending of the frame B, and the 
stake C, and that which may result from the elongation of the bolts 
which hold these members together. When the riveting pressure 
P is applied, the beams B and C tend to rotate around the point 
O, this tendency being resisted by the tension in the bolts. The 
load which may be applied to the bolts by the force P will be 

_ P (a + b) _. , , , t 1 

P t = . If the nuts on the bolts are set up so that a 

combined total initial tension somewhat greater than P x is induced 
in the bolts, the stretching of the bolts, and the consequent open- 
ing up between the frame and the stake, will be negligible. 



43 2 



MACHINE DESIGN 



(See Art. 60 and Fig. 43 and also Art. 77.) The intensity of 
stress in the bolts should not exceed 6,000 pounds per square inch. 
The upper part of the frame, B, approximates a cantilever of 
uniform strength of length a. (See Art. 15 and Case 1 of Table 
II.) The maximum deflection which occurs at D may, therefore, 
be computed and the maximum stress which occurs at E F may 
be checked by Equation J of Table VI. The stake, C, approxi- 
mates a cantilever of uniform cross-section, and may therefore be 




Fig. 185 



Fig. 186. 



treated in a similar manner. (See Case 1, Table I, and Equation 
/, Table VI.) 

Fig. 186 illustrates a closed frame as applied to a vertical 
steam engine. The back column, B, which carries the crosshead 
guide is of cast iron, while the front columns, C, are of steel. It 
is required to check the stresses in these columns when the piston 
is ascending and also when it is descending, the rotation of the 
engine to be taken in a clockwise direction as indicated. 

When the piston is ascending, the steam pressure tends to 
draw the cylinder and bed closer together. This tendency is 
resisted by P'., the combined thrust on all the columns, the vertical 
component of which must equal P, the total steam pressure on the 
piston. It may be reasonably assumed that the back column 



MACHINE FRAMES AND ATTACHMENTS 



433 



carries one-half of the total thrust, and that each of the front 

P' 
columns carries one-quarter. The thrust of the back column, — , 

2 

may be resolved into components perpendicular and parallel to 

p 

the face of the foot. The vertical component will equal — . The 

horizontal component R tends to spread the foot of the column 
outward and induce a bending stress in it. The column should, 
therefore, be secured to the bed by fitted bolts, or, if the bolts 
are loose in the holes, the foot should be well dowelled to the bed ; 
or, better still, the foot should fit against a ledge cast on the bed 
plate. R will then be balanced by an equal and opposite reaction 
at the feet of the front columns, thus setting up a negligible tension 
in the bed and leaving a compressive force only on the column. 
By similar reasoning each front column is subjected to a com- 

P' 
pressive load — and the total horizontal component R is balanced 

by that of the back column through the bed. 

The tension or compression in the piston-rod and connecting- 
rod, either ascending or descending, have a resultant R! normal 
to the guide, which may have a large value where the connecting- 
rod is short compared to the crank. This resultant tends to 
bend B, and hence C also, in a left-hand direction, the bending 
being resisted by the fastenings at the feet. The columns and 
cylinder, however, constitute a very stiff structure, and except 
where the frame is made up of light construction this effect may 
be neglected. This reaction, R', however, also bends the column 
B locally, that is as a beam encastre at S and N, the effect of R 
being greatest when the crosshead is near half stroke. (See 
Case 18, Table I.) If then it be desired to check the central 

P' 

section UV of the column, the long column stress due to — must 

2 

be added to the flexural stress due to R'. The sum of these 
stresses should not, of course, be greater than the allowable stress 
for the material used. The columns, C, need only be checked 
as long columns (see equation N, Table VI) . 



434 MACHINE DESIGN 

When the piston is descending the steam pressure tends to 
separate the bed and the cylinder. The reactions at M and N 
are reversed in direction and the columns are put in tension, the 
horizontal components inducing negligible compression in the 
bed. The most dangerous section in this case will be under R f 

P' 

and the stress will be that due to — plus the tensile stress due to 

2 

the bending effect of R'. The fastenings of the columns to the 
cylinder and to the bed plate must, of course, be sufficiently strong 
in tension to resist the force tending to separate the cylinder and 
bed. 

In the foregoing examples the lines of action of all forces 
acting on the section considered, lay in a plane of symmetry of 
the section, and the section tended to rotate around a gravity 
axis at right angles to this plane. While this is the most usual 
case, occasionally the force or forces acting are not in a plane of 
symmetry. Thus Fig. 187 may represent the cross-section of the 
column of a radial drilling machine, in which it is required to 
check the stresses when the force P, due to drilling, is in the 
position shown. If C be the centre of gravity of the section, the 
tendency to rotate will be around the axis X' X' at right angles to 
PC, the arm of the force P, and the resistance of the section against 
such rotation will be measured by the moment of inertia of the 
section with reference to this axis. The maximum tensile and 
compressive stresses will occur at the fibre farthest removed from 
X' X' or at M and N, the stress at M being tensile when the 
direction of P is upward to the plane of the paper, and com- 
pressive when its direction is downward. The centre of gravity, 
C, may be located readily, by finding the intersection of any pair 
of gravity axes. If the section has an axis of symmetry, as UV, 
Fig. 187, it is necessary only to find the axis at right angles to UV. 
This is most readily done graphically as follows: Divide the 
section into small areas, as shown by dotted lines at xx in Fig. 187. 
From the centre of gravity of each area draw parallel lines ab, 
be, cd, preferably at right angles to the known axis UV. In Fig. 
187 (a), lay off AB, BC, etc., proportional to the respective areas 



MACHINE FRAMES AND ATTACHMENTS 



435 



whose gravity axes are ab, be, etc. Take any pole O and draw 
AO, BO, etc. From any point on ab, draw ao indefinitely, parallel 
to AO. From the same point draw ob parallel to OB. From the 
intersection of ob and be draw co parallel to CO, and from its 
intersection with ed, draw od parallel to OD. The intersection of 
ao and od locates the gravity axis XX (see also Art. 120). It is 
evident that this method may be applied when both axes are un- 
known. 

The moment of inertia of the section around X' X f may be 
most readily found by transforming the area of the figure into 
an equivalent figure with RP as a base, as follows : Draw lines, 




Fig. 188 (b). 



as X" X" , parallel to X r X' ', and plot the intercepts made by it 
on the given section, on each side of CB as ordinates of an equi- 
valent section, shown in Fig. 187 by the dotted line L. The 
accuracy of the work may be checked with a planimeter, as it is 
evident that the area of the transformed section will be equal to 
that of the original. Divide this equivalent figure into approxi- 
mate rectangles, by lines drawn parallel to X r X r , as shown at 
r. Then the moment of inertia of r around the axis X' X' will 
be its moment of inertia round its own gravity axis parallel to 
X' X' ', plus its area into the square of the distance between these 
axes. The sum of the moments of inertia of all such small areas 
will be the required moment of inertia of the section. 



436 MACHINE DESIGN 

195. Distribution of Metal in Frames. Machine frames are 
usually made of castings, on account of their complicated shapes, 
cast iron being the material most used, while steel castings are rap- 
idly coming into use for severe work. In addition to the stresses 
induced in the frame by the energy transmitted by the machine, 
it may also be subjected to severe accidental stresses due to such 
causes as shrinkage, or the settling of a part of the foundation. 
Both these classes of stresses are, in general, very complex and gen- 
erally beyond mathematical analysis, and the problem must fre- 
quently be left to the judgment of the designer, especially if stiffness 
is a large factor. Economy in the use of metal, however, demands 
that its distribution throughout the frame shall be in accord 
with the best analysis possible, and, therefore, the general prin- 
ciples governing the forms of sections must be kept in mind. 

The most trying stresses to which a frame may be subjected 
are torsion, flexure, or a combination of these. It has been noted 
in Art. 12 that the hollow section (Fig. 7) is most effective for 
resisting torsion, and, if this be the predominating stress sections 
such as are shown in Fig. 7, or modified sections as shown in Fig. 
187, are correct. It was also noted in Art. 19 (Fig. 10) that in 
the case of cast iron, or other metal whose tensile strength is 
much less than its compressive strength, a great saving of material 
is effected by massing the metal on the tension side as shown in 
Fig. 188 (a) ; thus making the tensile and compressive stresses more 
in proportion to the strength of the material. If then the pre- 
dominating stress in a frame is simple flexure (in a given plane), 
a section like that shown in Fig. 188 (a) is allowable, but if, in 
addition, torsional strength must be withstood, or if the plane of 
flexure may change, a section similar to that shown in Fig. 188 (b) 
is better design, since it combines the merits of both Figs. 187 and 
188 (a). Sometimes it is better to make the section so large that 
the flexural stress can be safely withstood by a wall of uniform 
thickness, as in Fig. 187, as the construction of the pattern is 
simpler and the shrinkage stresses less serious than in such sec- 
tions as shown in Fig. 188. The metal in the walls will be much 
sounder, also, as the thick sections of Fig. 188 are very likely to 
have a porous interior, due to shrinkage. Cast-iron parts more 



MACHINE FRAMES AND ATTACHMENTS 437 

than four or five inches thick are almost sure to be defective in 
this manner. The walls of such sections as shown in Fig. 188 
should taper uniformly from the thick part to the thin parts, and 
all corners should be well rounded, and filleted, to minimize as 
far as possible the concentration of shrinkage stresses.- Thin 
wide flanges or webs should not be cast integral with thick heavy 
parts, as. unequal shrinkage and porosity are sure to result. This 
is especially true of thin ribs cast on the tension side of large 
sections, as the edge of the rib is liable to crack through shrinkage, 
thus starting rupture across the entire section. Small brackets 
or other attachments of thin sections should never be cast on a 
large frame, as they seldom cast well. A section of moderate 
thickness is often stronger than a thicker one, since the greatest 
strength of cast iron is in the outer skin. It should also be re- 
membered that even when a frame is both strong and stiff enough 
to do the required work at low speeds, it may not have mass 
enough to absorb the vibrations set up when running more 
rapidly. This may call for more metal in the frame than is 
dictated by other requirements. Openings for supporting or 
removing cores should be placed near the gravity axis so as 
to reduce the strength as little as possible (see Fig. 191). 

196. Attachments and Supports. The general appearance of 
a machine is affected more by the outline of the main frame than 
by that of any other member. This outline should, therefore, be 
clearly shown, and not obliterated at places by the various attach- 
ments which restrain the moving parts or support the frame. 
In Fig. 183 is shown the outline of a frame in which the various 
sections have been proportioned in accordance with the loads 
brought upon them, and the various bosses N and the support 5 
appear as attachments to the main member. Fig. 184 illustrates 
the same machine with the attachments merged into the main 
member, thereby destroying the character of the design, and also 
making it more difficult to judge of the relative strength of various 
sections of the frame. 

The form of an attachment will, of course, be governed by 
the service it is required to render and the manner in which it 
is loaded and supported. If the outline of the attachment is 



438 



MACHINE DESIGN 



based on theoretical considerations, care should be exercised 
that all the modifying influences are duly considered. Thus 
if parabolic outlines are given to an attachment, such as the 
housings H for supporting the tool in Fig. 192, the upper end of 
the housing must be modified from the theoretical parabolic 
outline indicated by the bending effect of the force P, so as to 
provide for the shearing effect at the upper end, which is fre 
quently neglected. (See also Article 15.) 

If the frame rests directly on the floor its outlines should be 
carried down to the floor in such a manner as will give an appear- 




Fig. 189. 



Fig. 190. 



ance of stability. Thus Fig. 189 shows such a machine frame 
on which the vertical outline of the back of the frame is undercut. 
Fig. 190 shows the same machine with the outline carried 
straight to the floor and the improvement in appearance, so far 
as stability is concerned, is obvious. Fig. 191 shows the outline 
of a planing machine in which the upright, U, is carried to the 
floor at V, in the form of a leg. This construction is not correct, 
as U is an attachment to the bed, designed to resist the force of 
the cut and transfer it to the bed, which should itself be stiff 
enough to withstand all such stress thus brought upon it. 
Any settling of the foundation might affect the alignment of 



MACHINE FRAMES AND ATTACHMENTS 



439 



U and hence the arrangement shown in Fig. 192 is more nearly 
correct. 

In large machines the frame usually rests directly on the 
foundation, and should have sufficient stiffness to resist distortion 
due to the settling of the foundation, since the latter is very 
difficult to avoid. In smaller machines the frame is carried on 
supports, which may be of two general types, (a) cabinet or box 
pillar supports (Fig. 192), and (b) legs as shown in Fig. 193. 
The choice of support will, of course, depend on the type and 
size of the machines. In any case the number of points of 
support should be as few as possible. If the machine can be 
supported on three points it is evident that the frame cannot be 
affected by settling of the foundation. It is difficult, in general, 




Fig. 192, 



to obtain three-point support, but it is seldom necessary to place 
supports as close together as in Fig. 191 (which is taken from an 
actual design), where the frame is carried on eight points. Fig. 
192 shows the same frame properly carried on box supports, 
the supports themselves being so stiff as materially to assist the 
frame and practically reducing the support to so-called two-point 
support. Small machines can often be supported on a single 
box-pillar, the overhanging parts of the frame having a parabolic 
outline as suggested in Fig. 189. If the box pillar is of 
considerable height the sides should taper slightly toward 
the top; for if made parallel the pillar will appear wider at 
the top than at the bottom. It is preferable to use one form 
of support throughout, i.e., all box pillars or all legs, and not 
one or more of each. 



440 



MACHINE DESIGN 



When the frame must be supported on legs, as in Fig. 195^ 
these should not curve outward as in Fig. 193, unless it is abso- 
lutely essential in order to obtain stability. Spreading the legs 
as in Fig. 193 lengthens the distance between the reactions, R, R, 
and, therefore, increases the bending effect on the bed and legs 
as a whole. The leg shown in profile in Fig. 194 is better and 
much easier to make. The legs should be so placed that the 
outline L forms a continuation of the principal vertical outline U 
of the frame, as shown in Fig. 194. The same remarks apply to 
the end view of the legs as shown in Figs. 195 and 196. The 
complex curves and ornate features of Fig. 195 are not only use- 





Fig. 193. 



Fig. 194. Fig. 195. 



Fig. 196. 



less but expensive. It is not always possible or desirable to make 
machine frames and supports with simple straight-line outlines; 
but where curves are necessary they should be as simple as 
possible ; and in general the best results can be obtained by using 
arcs of circles or parabolas. Ornamentation of a fanciful nature 
is not permissible anywhere, as it really detracts from the appear- 
ance of the machine, and adds to the cost of production. Har- 
mony of design can be attained by making the various members 
of correct proportions to withstand the loads brought upon 
them, and by using the simplest and most direct design with 
smooth transition curves between straight lines which intersect. 
It is a proverb in design that " what is right looks right." 



INDEX 



Absolute efficiency, no 
Accumulator, hydraulic, 29 
Air compressor, 26 
Air reservoir, 29 
Anti-friction metals, 233 
Apparent factor of safety, 89 
Axles, 285 

Babbitt metal, 233 
Ball bearings, 277 

allowable load on, 282 
Bands, thin, 205 

Barnard, Prof. W. N. (riveted fasten- 
ings), 149 
Beams, general theory of, 40 

of uniform strength, 42 
Bearing pressures on journals, table of, 

251 

on sliding surfaces, 238 
Bearing, step, 264 
Bearings, allowable pressure on, 251 

ball, 271, 277 

collar, 264 

construction of, 243 

forms of, 239 

metals for, 233 

perfectly lubricated, 253 

radiation of heat from, 247 

roller, 271, 273 

table of proportions of, 252 

thrust, 263 
Belt, example of design of, 314 

transmission, theory of, 308 
Belting, efficiency of, 318 

weight of, 314 
Belts, coefficient of friction of, 313 

construction of, 308 

creep of, 310 

practical consideration of, 319 

practical rules for, 318 



Belts, slip of, 310 

velocity of, 317 
Bending moment, equivalent or ideal, 49 
Bevel gears, 383 
Block brakes, 355 
Boiler plate, strength of, 154 

rivets, strength of, 154 
Bolts, allowable stress in, 176 

efficiency of, 172 

experiments on the strength of, 170 

for reinforcing castings, 207 

initial tension in, 169 

location of, 181 

Professor Sweet's experiments on, 
181 

resilience of, 178 

resultant stress in, 172 

straining action in, 168, 169, 172 

stud, 163, 164 

tap, 163, 164 

through, 163, 164 
Brakes, block, 355 

coefficient of friction for, 363 

differential, 359 

friction, 355 

strap, 357 
Briggs' system of pipe threads, 168 
Butt joints in plates, 138, 146 

Cap screws, 163 

Carman, Prof. A. P., experiments on 

tubes, 218 
Carrying strength, 89 
Chain drums and sheaves, 340 

Renold, Morse, 348 

roller, block, stud, 345 
Chains, 338 

conveyor, 344 

for power transmission, 344 

proof test of, 340 



441 



442 



INDEX 



Chains, silent, 345 

strength of, 340 

weldless, 340 
Clavarino's formula, 223 
Clutches, allowable pressure on, 363 

band, 362 

coefficient of friction for, 363 

conical, 359 

disc, 361 

friction, 359 

magnetic, 363 

radially expanding, 360 

shaft, 301, 305 
Coefficient of elasticity, 34 

of friction for screws, 184 
Coefficients of friction for brakes and 
clutches, 363 

of friction for friction wheels, 353 

of friction of pivots, 269 
Collar bearings, 264, 267 
Columns, eccentric loading of, 73 

or long struts, 61 
Compression and torsion, combined, 57 

in machine elements, 36 
Conservation of energy, 3-6 
Constraining surfaces, materials of, 232 
Continuous system of rope-driving, 329 
Cotters, stresses in, 198 
Coupling, flange shaft, 303 

Hook's, 304 

Oldham, 304 
Couplings, flexible shaft, 306 

shaft, 301 
Crank-effort diagram, 20 
Cycloidal gear teeth, 365 
Cylinders, thick, 223 

thin, 2ir, 215 

Deflection of ropes, 332 

table of, 333 
Deformation, work of, 77 
Differential brake, 359 
Discs, rotating, 425 

Efficiency, absolute, no 
definition of, 6 
general theory of, 109 
mechanical, no 



Efficiency of belting, 318 

of bolts, 172 

of riveted fastenings, 141 

of screws, 157 

of square- threaded screws, 157 

of triangular-threaded screws, 162 
Efficiencies of machine elements, 113 
Elastic limit, 34 

resilience, 77 
Elasticity, coefficient of, 34 
Energy cycle, 6 

in air compressor, 26 

in steam engine, 16 
Energy problems, 6 

redistribution of, 29 
Euler's formula for columns, 62 

Factor of safety, 35, 88 

on boiler work, 155 
Factors of safety, table of, 91 
Fairbairn, Sir Wm., experiments on 

flues, 217 
Fatigue of materials, 82 
Feather keys, 196 

Feathers, table of dimensions of, 197 
Flanges, pipe, 226 

Flather, Prof., on rope drives, 323, 324 
Flexure and direct stress, 58 
torsion combined, 43 

in machine elements, 40 
Flues, 211 

Flywheel rim joints, 422 
Flywheels, 406 

coefficients of fluctuation, 412 

construction of, 419 

experiments on the strength of, 424 

general theory of, 406 

stresses in, 413 
Force fits, 200 

practical considerations in, 204 
stresses due to, 201 
Forces acting on machines, 6, 9, 31 
Friction, applications of, 350 

clutches, 359 

coefficient of, 97, 99, 104, 105 

general theory of, 96 

laws of, 98 

of circular surfaces, 97 



INDEX 



443 



Friction of dry surfaces, 98 
of flat surfaces, 97 
of lubricated surfaces, 99 
of triangular threads, 162 
of screws, 157 
of rolling, 99 
static, 100 

summary of general laws of, 109 
wheels, allowable pressures on, 352 

coefficients of friction for, 353 

forms of, 350 

materials for, 352 

power transmitted by, 353 

wedge-faced, 354 
work of, 97 
Furnace flues, corrugated, 222 

Gear teeth, allowable stresses in, 386 

cut, 369 

cycloidal, 365 

Fellows system of stub, 390 

Hunt system, 390 

involute, 365 

machine moulded, 369 

methods of making, 369 

proportions of, 368, 370 

shrouding of, 389 

strength of, 376 

stub, 390 

wear on, 388 

width of face of, 388 
wheels, forces acting on, 373 

mortise, 371 

rawhide, 372 

strength of rims and arms, 391 
Gearing, efficiency of spur, 392 
general principles of, 364 
' helical or twisted, 392 
herring-bone, 393 
interchangeable systems of, 366 
screw, 395 " 
skew-bevel, 395 
spiral, 395 

standard forms of, 366 
strength of twisted, 393 
worm, 395 
Gears, allowable speed of, 387 
bevel, 383 



Gears, rawhide, allowable load on, 389 
Gordon's formula for columns, 67 

Helical gearing, 392 
Hindley worm, 398 
Hobs and hobbing, 397 
Hoisting mechanism, 9, 29 
Hook's coupling, 305 
Hooks, hoisting, 341 

strength of, 341 

table of proportions of, 343 
Hoops, 205 

Hunt, C. W., on rope driving, 325 
system of gear teeth, 390 

Impact, shock, 78 
Imperfect lubrication, 10a 
Inertia effects in general, 29 
redistribution of, 29 
Inertia forces in steam engines, 19 
Involute gear teeth, 365 

Johnson's, J. B., formula for columns, 
66 
T. H., formula for columns, 65 

Journals, bearing pressure on, 251 
design of, 245, 257 
examples of design of, 257 
imperfectly lubricated, 249 
perfectly lubricated, 253 

Keys, draw, 192 

flat, 190 

forms of, 190 

saddle, 190 

stresses in, 192 

sunk, 190 

table of dimensions of sunk, 196 

Woodruff, 191 
Kinematics, 6 

Lame's formula, 223 

Lap joints in plates, 138, 145 

Lasche, experiments of, 247, 255 

Launhart's formula, 85 

Lewis', Wilfred, formula for gear teeth, 

376 
Live load, effect of, 82 



444 



INDEX 



Load, steady, dead, suddenly applied, 31 
Lubrication, imperfect, 102 

methods of, 100, 261 

of journals, methods of, 261 

of sliding surfaces, 238 

perfect, 106 

Machine attachments, 437 
design, definition of, 1 
frames, 428 

distribution of metal in, 436 
stresses in, 428 
stresses in closed, 432 
stresses in open, 429 
screws, 163, 164 
supports, 437 
McBride, James, experiments on effi- 
ciency of bolts, 172 
Mechanical advantage, 28 

efficiency, no 
Mechanism, definition of, 2 
Micro flaws, theory of, 83 
Moore, Prof. H. F., experiments of, 107 

on riveted fastenings, 149 
Morse chain, 348 
Multiple system of rope-driving, 329 

Oil film, 106 

in perfect lubrication, 106 
grooves, 262 
Oldham coupling, 304 

Perfect lubrication, 106 
Pipe couplings and flanges, 226 

threads, 168 
Pipes, 211, 217 

Piping, practical considerations of, 224 
Pivots, coefficient of friction of, 269 
.Planing machine, 30 
Plates, thin, 228 
Power, definition of, 7 
Pulleys, 406 
Punching machine, 10, 60 

Rankine's equation for columns, 67 
Relative strength of riveted fastenings, 

141 
Renold chain, 348 



Resilience, 76 
elastic, 77 
of bolts, 178 
Ritter's formula for columns, 68 
Riveted fastenings, butt joints, 138, 146 
chain riveting, 138 
efficiency of, 141 
factor of safety in, 155 
failure of, 141 
forms of joints, 137 
general considerations, 136 
general equations for, 147 
lap joints, 138, 144 
making of, 151 
marginal strength of, 143 
practical consideration of, 149 
practical rules for, 155 
relative strength of, 141 
staggered riveting, 138 
strength of materials foi, 154 
stresses in, 139 
theoretical strength of, 144 
Riveting, machine, 153 
Rivets, diagonal pitch of, 138, 143 
pitch of, 138 

transverse pitch of, 138, 143 
Roller bearings, 271, 273 

allowable load on, 276 
Rope-driving, sheaves for fibrous, 331 

systems of fibrous, 329 
Rope transmission, theory of, 309, 323 

(by wire), 338 
Ropes, cotton, 322 
Ropes, deflection of fibrous, 332 
fibrous hoisting, 337 
hemp, leather, etc., 322 
Manila, 322 

materials for fibrous, 322 
materials for wire, 334 
strength of fibrous, 327 
strength of fibrous hoisting, 338 
strength of wire hoisting, 339 
velocity of fibrous, 327 
wire hoisting ropes, 338 
Rotating discs, 425 

Screw fastenings, 163 
gearing, 395 



INDEX 



445 



Screw and screw fastenings, 156 
Screws, bearing pressure on, 186 

cap, 163, 164 

coefficient of friction of, 184 

design of, for power transmission, 
187 

efficiency of, 157 

for power transmission, 183 

for power transmission, efficiency 
of, 184 

forms of, 156 

friction of, 157 

machine, 163, 164 

mechanical advantage of, 183 

multiple- threaded, 184 

stresses in transmission, 186 

U. S. or Sellers standard, 166 

Whitworth standard, 166 
Sellers shaft coupling, 303 

standard screws, 166 
Set screws, 163, 165 
Shaft clutches, 301 

coupling, flange, 303 

couplings, 301 
Shafts, allowable deflection of, 299 

allowable span of, 299 

factors of safety for, 289 

hollow, 300 

subjected to torsion, 288 

subjected to torsion and bending, 
290 

torsional stiffness of, 298 

whirling of, 299 
Shaping machine, energy distribution 

in, 10 
Shear in machine elements, 36 
Shock in machine members, 78 
Shrink fits, 200, 207 

practical considerations in, 204 
Shrouding of gear teeth, 389 
Sliding surfaces, 233 

bearing pressures on, 238 
lubrication of, 238 
Spheres, 213 
Splines, 196 
Springs, applications of, 114 

characteristics of, 114 

flat, 116 



Springs, flat, design of, 119 
forms of, 116 
helical, 117 

design of, 128 
springs in torsion, 135 
spiral, 118 
laminated or plate, design of, 123 
materials of, 115 
spiral, 118 
Spur gear teeth, strength of, 376 
gearing, efficiency of, 392 
gears, allowable speed of, 387 
allowable stress in, 386 
machine-moulded, 369 
width of face of, 388 
Stayed surfaces, 231 
Steam engine, energy distribution in, 16 
Step bearing, 264 
Stewart, Prof. R. T., experiments on 

tubes, 218 
Storage battery, 29 
Strain, definition of, 32 
Straining action, nature of, 32 

table pf formula? for, 94 
Strap brakes, 357 
Strength of materials, table of, 93 
Stress, compound, 33, 40 
definition of, 32 
predominating or primary, 40 
strain diagram, 33 
working, 35 
Stribeck, Prof., experiments of, 255, 272 
Stub gear teeth, 390 
Stud bolts, 163, 164 
Sweet, Prof., method of relieving sliding 
surfaces, 237 

Tap bolts, 164 

Taylor, F. M., rules for belting, 319 

Temperature, coefficient of expansion, 

75 
stresses due to, 75 
Tension in machine elements, 35 
Thrust bearing for worms, 405 
Thrust bearings, 263 

allowable pressures on, 270 
efficiency of, 268 
Toothed gearing, angular velocity of, 365 



446 



INDEX 



Toothed gearing, classification of, 364 
interchangeable systems of, 
366 
Torsion and compression, combined, 

57 
and flexure, combined, 43 

in machine elements, 36 

Tower, Beaucamp, experiments of, 254 

Tower's experiments, 106 

Towne, H. R., experiments on hooks, 

343 
Triangular threads, efficiency of, 162 

friction of, 162 

Tubes, 2ii, 217 

Twisted gears, 392 

Ultimate strength, definition of, 34 

Unions, pipe, 226 

U- S. standard screws, table of, 167 



Van Stone pipe flanges, 227 

Weyrauch's formula, 86 
Whit worth standard screws, 166 
Wire rope transmission, ^^^ 
theory of, 334 
ropes, materials for, 334 

power transmitted by, 336 
Wohler's experiments of, 84 
Work of deformation, 76 
Working stress, 35 
Worm and worm wheel, 395 
gearing, design of, 403 
efficiency of, 399 
limiting pressures on, 401 
limiting velocities of, 401 
velocity ratio of, 398 
Hindley, 398 
thrust bearing, 405 




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